Question
1) Verify whether the below two functions are one-to-one and/or onto.
Let f : R+ -> R be defined by f(x) = 3*x*x
Let function g : R - {0} -> R defined by g(x) = 1/(3x).

2) What must be true in order for a function f to have an inverse function?
A- All functions have inverses.
B- f must be onto but not one-to-one
C- f must be one-to-one but not onto.
D- f must be a one-to-one correspondence.

3) Let A = {1,2,3} and B = {4,5,6,7}. Let f = {(1,5), (2,4),(3,6)}.
Select the correct statement:
A) Inverse of f = {(5,2),(4,1),(6,3)}
B) f does not have an inverse.
C) Inverse of f = {(5,1),(4,2),(6,3)}
D) Inverse of f = {(5,5),(1,1),(3,3)}
Solution Preview

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references.
Students may use these solutions for personal skill-building and practice.
Unethical use is strictly forbidden.

This is only a preview of the solution.
Please use the purchase button to see the entire solution.
By purchasing this solution you'll be able to access the following files:
Solution.docx
Purchase Solution
$5.00
Google Pay
Amazon
Paypal
Mastercard
Visacard
Discover
Amex
View Available Computer Science Tutors 529 tutors matched
ionut
Ionut
(ionut)
Hi! MSc Applied Informatics & Computer Science Engineer. Practical experience in many CS & IT branches.Research work & homework
5/5 (5,654+ sessions)
2 hours avg response
Leo
(Leo)
Hi! I have been a professor in New York and taught in a math department and in an applied math department.
4.9/5 (5,652+ sessions)
2 hours avg response
Pranay
(math1983)
Ph.D. in mathematics and working as an Assistant Professor in University. I can provide help in mathematics, statistics and allied areas.
4.6/5 (5,512+ sessions)
1 hour avg response

Similar Homework Solutions

8.1.0PHP Version439msRequest Duration45MBMemory UsageGET college-homework-library/{category}/{subject}/{id}Route
    • Booting (281ms)time
    • Application (158ms)time
    • 1 x Booting (64.09%)
      281ms
      1 x Application (35.91%)
      158ms
      • Illuminate\Routing\Events\Routing (1.37ms)
      • Illuminate\Routing\Events\RouteMatched (559μs)
      • Illuminate\Foundation\Events\LocaleUpdated (5.82ms)
      • eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (252μs)
      • eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (333μs)
      • Illuminate\Database\Events\ConnectionEstablished (1.38ms)
      • Illuminate\Database\Events\StatementPrepared (14.77ms)
      • Illuminate\Database\Events\QueryExecuted (1.41ms)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (106μs)
      • eloquent.booting: App\Models\Subject (5.28ms)
      • eloquent.booted: App\Models\Subject (64μs)
      • Illuminate\Database\Events\StatementPrepared (1.52ms)
      • Illuminate\Database\Events\QueryExecuted (841μs)
      • eloquent.retrieved: App\Models\Subject (1.6ms)
      • eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibraryFile (117μs)
      • eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibraryFile (47μs)
      • Illuminate\Database\Events\StatementPrepared (664μs)
      • Illuminate\Database\Events\QueryExecuted (916μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (68μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (15μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (8μs)
      • eloquent.booting: App\Models\SubjectCat (491μs)
      • eloquent.booted: App\Models\SubjectCat (40μs)
      • Illuminate\Database\Events\StatementPrepared (764μs)
      • Illuminate\Database\Events\QueryExecuted (1.11ms)
      • eloquent.retrieved: App\Models\SubjectCat (99μs)
      • Illuminate\Cache\Events\CacheHit (14.87ms)
      • Illuminate\Cache\Events\CacheMissed (339μs)
      • Illuminate\Database\Events\StatementPrepared (1.02ms)
      • Illuminate\Database\Events\QueryExecuted (7.75ms)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (106μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (17μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (11μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (9μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (7μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (6μs)
      • Illuminate\Database\Events\StatementPrepared (788μs)
      • Illuminate\Database\Events\QueryExecuted (1.15ms)
      • eloquent.retrieved: App\Models\Subject (97μs)
      • Illuminate\Cache\Events\KeyWritten (1.46ms)
      • Illuminate\Database\Events\StatementPrepared (2.06ms)
      • Illuminate\Database\Events\QueryExecuted (1.16ms)
      • Illuminate\Database\Events\StatementPrepared (1.07ms)
      • Illuminate\Database\Events\QueryExecuted (1.08ms)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (56μs)
      • Illuminate\Database\Events\StatementPrepared (758μs)
      • Illuminate\Database\Events\QueryExecuted (1.07ms)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (54μs)
      • Illuminate\Cache\Events\CacheHit (960μs)
      • creating: homework.show (396μs)
      • composing: homework.show (197μs)
      • creating: components.breadcrumbs (370μs)
      • composing: components.breadcrumbs (188μs)
      • Illuminate\Database\Events\StatementPrepared (1.41ms)
      • Illuminate\Database\Events\QueryExecuted (1.28ms)
      • eloquent.retrieved: App\Models\SubjectCat (112μs)
      • Illuminate\Cache\Events\CacheHit (4.3ms)
      • Illuminate\Cache\Events\CacheHit (224μs)
      • Illuminate\Cache\Events\CacheHit (221μs)
      • Illuminate\Cache\Events\CacheHit (174μs)
      • Illuminate\Cache\Events\CacheHit (200μs)
      • Illuminate\Cache\Events\CacheHit (168μs)
      • Illuminate\Cache\Events\CacheHit (207μs)
      • Illuminate\Cache\Events\CacheHit (172μs)
      • Illuminate\Cache\Events\CacheHit (191μs)
      • Illuminate\Cache\Events\CacheHit (168μs)
      • Illuminate\Cache\Events\CacheHit (197μs)
      • Illuminate\Cache\Events\CacheHit (167μs)
      • Illuminate\Cache\Events\CacheHit (193μs)
      • Illuminate\Cache\Events\CacheHit (166μs)
      • Illuminate\Cache\Events\CacheHit (249μs)
      • Illuminate\Cache\Events\CacheHit (205μs)
      • Illuminate\Cache\Events\CacheHit (233μs)
      • Illuminate\Cache\Events\CacheHit (245μs)
      • Illuminate\Cache\Events\CacheHit (228μs)
      • Illuminate\Cache\Events\CacheHit (189μs)
      • Illuminate\Cache\Events\CacheHit (214μs)
      • Illuminate\Cache\Events\CacheHit (185μs)
      • Illuminate\Cache\Events\CacheHit (223μs)
      • Illuminate\Cache\Events\CacheHit (169μs)
      • Illuminate\Cache\Events\CacheHit (196μs)
      • Illuminate\Cache\Events\CacheHit (170μs)
      • Illuminate\Cache\Events\CacheHit (199μs)
      • Illuminate\Cache\Events\CacheHit (163μs)
      • Illuminate\Cache\Events\CacheHit (214μs)
      • Illuminate\Cache\Events\CacheHit (188μs)
      • Illuminate\Cache\Events\CacheHit (211μs)
      • Illuminate\Cache\Events\CacheHit (190μs)
      • Illuminate\Cache\Events\CacheHit (208μs)
      • Illuminate\Cache\Events\CacheHit (169μs)
      • Illuminate\Cache\Events\CacheHit (221μs)
      • Illuminate\Cache\Events\CacheHit (188μs)
      • Illuminate\Cache\Events\CacheHit (221μs)
      • Illuminate\Cache\Events\CacheHit (284μs)
      • Illuminate\Cache\Events\CacheHit (201μs)
      • Illuminate\Cache\Events\CacheHit (168μs)
      • Illuminate\Cache\Events\CacheHit (191μs)
      • Illuminate\Cache\Events\CacheHit (164μs)
      • Illuminate\Cache\Events\CacheHit (187μs)
      • Illuminate\Cache\Events\CacheHit (162μs)
      • Illuminate\Cache\Events\CacheHit (215μs)
      • Illuminate\Cache\Events\CacheHit (189μs)
      • Illuminate\Cache\Events\CacheHit (202μs)
      • Illuminate\Cache\Events\CacheHit (167μs)
      • Illuminate\Cache\Events\CacheHit (212μs)
      • Illuminate\Cache\Events\CacheHit (193μs)
      • Illuminate\Cache\Events\CacheHit (305μs)
      • Illuminate\Cache\Events\CacheHit (260μs)
      • Illuminate\Cache\Events\CacheHit (224μs)
      • Illuminate\Cache\Events\CacheHit (186μs)
      • Illuminate\Cache\Events\CacheHit (221μs)
      • Illuminate\Cache\Events\CacheHit (188μs)
      • Illuminate\Cache\Events\CacheHit (249μs)
      • Illuminate\Cache\Events\CacheHit (182μs)
      • Illuminate\Cache\Events\CacheHit (199μs)
      • Illuminate\Cache\Events\CacheHit (166μs)
      • Illuminate\Cache\Events\CacheHit (192μs)
      • Illuminate\Cache\Events\CacheHit (164μs)
      • Illuminate\Cache\Events\CacheHit (214μs)
      • Illuminate\Cache\Events\CacheHit (192μs)
      • Illuminate\Cache\Events\CacheHit (216μs)
      • Illuminate\Cache\Events\CacheHit (187μs)
      • Illuminate\Cache\Events\CacheHit (210μs)
      • Illuminate\Cache\Events\CacheHit (186μs)
      • Illuminate\Cache\Events\CacheHit (214μs)
      • Illuminate\Cache\Events\CacheHit (185μs)
      • Illuminate\Cache\Events\CacheHit (210μs)
      • Illuminate\Cache\Events\CacheHit (187μs)
      • Illuminate\Cache\Events\CacheHit (211μs)
      • Illuminate\Cache\Events\CacheHit (187μs)
      • Illuminate\Cache\Events\CacheHit (210μs)
      • Illuminate\Cache\Events\CacheHit (185μs)
      • Illuminate\Cache\Events\CacheHit (282μs)
      • Illuminate\Cache\Events\CacheHit (198μs)
      • Illuminate\Cache\Events\CacheHit (214μs)
      • Illuminate\Cache\Events\CacheHit (186μs)
      • Illuminate\Cache\Events\CacheHit (214μs)
      • Illuminate\Cache\Events\CacheHit (189μs)
      • Illuminate\Cache\Events\CacheHit (218μs)
      • Illuminate\Cache\Events\CacheHit (187μs)
      • Illuminate\Cache\Events\CacheHit (208μs)
      • Illuminate\Cache\Events\CacheHit (186μs)
      • Illuminate\Cache\Events\CacheHit (209μs)
      • Illuminate\Cache\Events\CacheHit (184μs)
      • Illuminate\Cache\Events\CacheHit (206μs)
      • Illuminate\Cache\Events\CacheHit (167μs)
      • Illuminate\Cache\Events\CacheHit (199μs)
      • Illuminate\Cache\Events\CacheHit (167μs)
      • Illuminate\Cache\Events\CacheHit (272μs)
      • Illuminate\Cache\Events\CacheHit (195μs)
      • Illuminate\Cache\Events\CacheHit (218μs)
      • Illuminate\Cache\Events\CacheHit (176μs)
      • Illuminate\Cache\Events\CacheHit (232μs)
      • Illuminate\Cache\Events\CacheHit (172μs)
      • Illuminate\Cache\Events\CacheHit (183μs)
      • Illuminate\Cache\Events\CacheHit (165μs)
      • Illuminate\Cache\Events\CacheHit (188μs)
      • Illuminate\Cache\Events\CacheHit (159μs)
      • Illuminate\Cache\Events\CacheHit (213μs)
      • Illuminate\Cache\Events\CacheHit (168μs)
      • Illuminate\Cache\Events\CacheHit (196μs)
      • Illuminate\Cache\Events\CacheHit (163μs)
      • Illuminate\Cache\Events\CacheHit (225μs)
      • Illuminate\Cache\Events\CacheHit (195μs)
      • Illuminate\Cache\Events\CacheHit (212μs)
      • Illuminate\Cache\Events\CacheHit (186μs)
      • Illuminate\Cache\Events\CacheHit (199μs)
      • Illuminate\Cache\Events\CacheHit (161μs)
      • Illuminate\Cache\Events\CacheHit (210μs)
      • Illuminate\Cache\Events\CacheHit (188μs)
      • Illuminate\Cache\Events\CacheHit (962μs)
      • Illuminate\Cache\Events\CacheHit (171μs)
      • Illuminate\Cache\Events\CacheHit (189μs)
      • Illuminate\Cache\Events\CacheHit (160μs)
      • Illuminate\Cache\Events\CacheHit (187μs)
      • Illuminate\Cache\Events\CacheHit (167μs)
      • Illuminate\Cache\Events\CacheHit (187μs)
      • Illuminate\Cache\Events\CacheHit (165μs)
      • Illuminate\Cache\Events\CacheHit (193μs)
      • Illuminate\Cache\Events\CacheHit (163μs)
      • Illuminate\Cache\Events\CacheHit (186μs)
      • Illuminate\Cache\Events\CacheHit (162μs)
      • Illuminate\Cache\Events\CacheHit (186μs)
      • Illuminate\Cache\Events\CacheHit (166μs)
      • Illuminate\Cache\Events\CacheHit (184μs)
      • Illuminate\Cache\Events\CacheHit (162μs)
      • Illuminate\Cache\Events\CacheHit (227μs)
      • Illuminate\Cache\Events\CacheHit (192μs)
      • Illuminate\Cache\Events\CacheHit (213μs)
      • Illuminate\Cache\Events\CacheHit (159μs)
      • Illuminate\Cache\Events\CacheHit (195μs)
      • Illuminate\Cache\Events\CacheHit (194μs)
      • Illuminate\Cache\Events\CacheHit (194μs)
      • Illuminate\Cache\Events\CacheHit (175μs)
      • Illuminate\Cache\Events\CacheHit (172μs)
      • Illuminate\Cache\Events\CacheHit (155μs)
      • Illuminate\Cache\Events\CacheHit (192μs)
      • Illuminate\Cache\Events\CacheHit (165μs)
      • Illuminate\Cache\Events\CacheHit (186μs)
      • Illuminate\Cache\Events\CacheHit (162μs)
      • Illuminate\Cache\Events\CacheHit (185μs)
      • Illuminate\Cache\Events\CacheHit (162μs)
      • Illuminate\Cache\Events\CacheHit (182μs)
      • Illuminate\Cache\Events\CacheHit (161μs)
      • Illuminate\Cache\Events\CacheHit (184μs)
      • Illuminate\Cache\Events\CacheHit (160μs)
      • Illuminate\Cache\Events\CacheHit (184μs)
      • Illuminate\Cache\Events\CacheHit (160μs)
      • Illuminate\Cache\Events\CacheHit (183μs)
      • Illuminate\Cache\Events\CacheHit (160μs)
      • Illuminate\Cache\Events\CacheHit (192μs)
      • Illuminate\Cache\Events\CacheHit (160μs)
      • Illuminate\Cache\Events\CacheHit (183μs)
      • Illuminate\Cache\Events\CacheHit (178μs)
      • Illuminate\Cache\Events\CacheHit (260μs)
      • Illuminate\Cache\Events\CacheHit (172μs)
      • Illuminate\Cache\Events\CacheHit (187μs)
      • Illuminate\Cache\Events\CacheHit (164μs)
      • Illuminate\Cache\Events\CacheHit (236μs)
      • Illuminate\Cache\Events\CacheHit (200μs)
      • Illuminate\Cache\Events\CacheHit (212μs)
      • Illuminate\Cache\Events\CacheHit (188μs)
      • Illuminate\Cache\Events\CacheHit (202μs)
      • Illuminate\Cache\Events\CacheHit (166μs)
      • Illuminate\Cache\Events\CacheHit (187μs)
      • Illuminate\Cache\Events\CacheHit (162μs)
      • Illuminate\Cache\Events\CacheHit (185μs)
      • Illuminate\Cache\Events\CacheHit (161μs)
      • Illuminate\Cache\Events\CacheHit (185μs)
      • Illuminate\Cache\Events\CacheHit (166μs)
      • Illuminate\Cache\Events\CacheHit (185μs)
      • Illuminate\Cache\Events\CacheHit (162μs)
      • Illuminate\Cache\Events\CacheHit (183μs)
      • Illuminate\Cache\Events\CacheHit (161μs)
      • Illuminate\Cache\Events\CacheHit (184μs)
      • Illuminate\Cache\Events\CacheHit (182μs)
      • Illuminate\Cache\Events\CacheHit (238μs)
      • Illuminate\Cache\Events\CacheHit (176μs)
      • Illuminate\Cache\Events\CacheHit (239μs)
      • Illuminate\Cache\Events\CacheHit (193μs)
      • Illuminate\Cache\Events\CacheHit (211μs)
      • Illuminate\Cache\Events\CacheHit (185μs)
      • Illuminate\Cache\Events\CacheHit (200μs)
      • Illuminate\Cache\Events\CacheHit (163μs)
      • Illuminate\Cache\Events\CacheHit (185μs)
      • Illuminate\Cache\Events\CacheHit (162μs)
      • Illuminate\Cache\Events\CacheHit (192μs)
      • Illuminate\Cache\Events\CacheHit (155μs)
      • Illuminate\Cache\Events\CacheHit (177μs)
      • Illuminate\Cache\Events\CacheHit (153μs)
      • Illuminate\Cache\Events\CacheHit (188μs)
      • Illuminate\Cache\Events\CacheHit (162μs)
      • Illuminate\Cache\Events\CacheHit (215μs)
      • Illuminate\Cache\Events\CacheHit (176μs)
      • Illuminate\Cache\Events\CacheHit (201μs)
      • Illuminate\Cache\Events\CacheHit (169μs)
      • Illuminate\Cache\Events\CacheHit (239μs)
      • Illuminate\Cache\Events\CacheHit (268μs)
      • Illuminate\Cache\Events\CacheHit (207μs)
      • Illuminate\Cache\Events\CacheHit (166μs)
      • Illuminate\Cache\Events\CacheHit (257μs)
      • Illuminate\Cache\Events\CacheHit (166μs)
      • Illuminate\Cache\Events\CacheHit (188μs)
      • Illuminate\Cache\Events\CacheHit (170μs)
      • Illuminate\Cache\Events\CacheHit (185μs)
      • Illuminate\Cache\Events\CacheHit (161μs)
      • Illuminate\Cache\Events\CacheHit (184μs)
      • Illuminate\Cache\Events\CacheHit (163μs)
      • Illuminate\Cache\Events\CacheHit (211μs)
      • Illuminate\Cache\Events\CacheHit (178μs)
      • Illuminate\Cache\Events\CacheHit (990μs)
      • Illuminate\Cache\Events\CacheHit (178μs)
      • Illuminate\Cache\Events\CacheHit (195μs)
      • Illuminate\Cache\Events\CacheHit (222μs)
      • Illuminate\Cache\Events\CacheHit (207μs)
      • Illuminate\Cache\Events\CacheHit (169μs)
      • Illuminate\Cache\Events\CacheHit (189μs)
      • Illuminate\Cache\Events\CacheHit (165μs)
      • Illuminate\Cache\Events\CacheHit (188μs)
      • Illuminate\Cache\Events\CacheHit (169μs)
      • Illuminate\Cache\Events\CacheHit (187μs)
      • Illuminate\Cache\Events\CacheHit (165μs)
      • Illuminate\Cache\Events\CacheHit (185μs)
      • Illuminate\Cache\Events\CacheHit (163μs)
      • Illuminate\Cache\Events\CacheHit (189μs)
      • Illuminate\Cache\Events\CacheHit (164μs)
      • Illuminate\Cache\Events\CacheHit (202μs)
      • Illuminate\Cache\Events\CacheHit (156μs)
      • Illuminate\Cache\Events\CacheHit (237μs)
      • Illuminate\Cache\Events\CacheHit (192μs)
      • Illuminate\Cache\Events\CacheHit (211μs)
      • Illuminate\Cache\Events\CacheHit (167μs)
      • Illuminate\Cache\Events\CacheHit (193μs)
      • Illuminate\Cache\Events\CacheHit (164μs)
      • Illuminate\Cache\Events\CacheHit (203μs)
      • Illuminate\Cache\Events\CacheHit (232μs)
      • Illuminate\Cache\Events\CacheHit (202μs)
      • Illuminate\Cache\Events\CacheHit (165μs)
      • Illuminate\Cache\Events\CacheHit (187μs)
      • Illuminate\Cache\Events\CacheHit (164μs)
      • Illuminate\Cache\Events\CacheHit (185μs)
      • Illuminate\Cache\Events\CacheHit (164μs)
      • Illuminate\Cache\Events\CacheHit (185μs)
      • Illuminate\Cache\Events\CacheHit (163μs)
      • Illuminate\Cache\Events\CacheHit (185μs)
      • Illuminate\Cache\Events\CacheHit (163μs)
      • Illuminate\Cache\Events\CacheHit (185μs)
      • Illuminate\Cache\Events\CacheHit (163μs)
      • Illuminate\Cache\Events\CacheHit (203μs)
      • Illuminate\Cache\Events\CacheHit (172μs)
      • Illuminate\Cache\Events\CacheHit (194μs)
      • Illuminate\Cache\Events\CacheHit (165μs)
      • Illuminate\Cache\Events\CacheHit (188μs)
      • Illuminate\Cache\Events\CacheHit (164μs)
      • Illuminate\Cache\Events\CacheHit (184μs)
      • Illuminate\Cache\Events\CacheHit (164μs)
      • Illuminate\Cache\Events\CacheHit (355μs)
      • Illuminate\Cache\Events\CacheHit (175μs)
      • Illuminate\Cache\Events\CacheHit (184μs)
      • Illuminate\Cache\Events\CacheHit (158μs)
      • Illuminate\Cache\Events\CacheHit (180μs)
      • Illuminate\Cache\Events\CacheHit (160μs)
      • Illuminate\Cache\Events\CacheHit (223μs)
      • Illuminate\Cache\Events\CacheHit (182μs)
      • Illuminate\Cache\Events\CacheHit (194μs)
      • Illuminate\Cache\Events\CacheHit (164μs)
      • Illuminate\Cache\Events\CacheHit (190μs)
      • Illuminate\Cache\Events\CacheHit (161μs)
      • Illuminate\Cache\Events\CacheHit (179μs)
      • Illuminate\Cache\Events\CacheHit (160μs)
      • Illuminate\Cache\Events\CacheHit (182μs)
      • Illuminate\Cache\Events\CacheHit (166μs)
      • Illuminate\Cache\Events\CacheHit (184μs)
      • Illuminate\Cache\Events\CacheHit (153μs)
      • Illuminate\Cache\Events\CacheHit (204μs)
      • Illuminate\Cache\Events\CacheHit (161μs)
      • Illuminate\Cache\Events\CacheHit (180μs)
      • Illuminate\Cache\Events\CacheHit (159μs)
      • Illuminate\Cache\Events\CacheHit (212μs)
      • Illuminate\Cache\Events\CacheHit (241μs)
      • Illuminate\Cache\Events\CacheHit (183μs)
      • Illuminate\Cache\Events\CacheHit (156μs)
      • Illuminate\Cache\Events\CacheHit (186μs)
      • Illuminate\Cache\Events\CacheHit (152μs)
      • Illuminate\Cache\Events\CacheHit (180μs)
      • Illuminate\Cache\Events\CacheHit (171μs)
      • Illuminate\Cache\Events\CacheHit (185μs)
      • Illuminate\Cache\Events\CacheHit (160μs)
      • Illuminate\Cache\Events\CacheHit (193μs)
      • Illuminate\Cache\Events\CacheHit (166μs)
      • Illuminate\Cache\Events\CacheHit (187μs)
      • Illuminate\Cache\Events\CacheHit (164μs)
      • Illuminate\Cache\Events\CacheHit (185μs)
      • Illuminate\Cache\Events\CacheHit (159μs)
      • Illuminate\Cache\Events\CacheHit (176μs)
      • Illuminate\Cache\Events\CacheHit (159μs)
      • Illuminate\Cache\Events\CacheHit (174μs)
      • Illuminate\Cache\Events\CacheHit (156μs)
      • Illuminate\Cache\Events\CacheHit (177μs)
      • Illuminate\Cache\Events\CacheHit (151μs)
      • creating: site.layouts.app (437μs)
      • composing: site.layouts.app (21μs)
      • creating: components.canonical (420μs)
      • composing: components.canonical (107μs)
      • creating: components.open-graph (203μs)
      • composing: components.open-graph (78μs)
      • creating: site.headers.header (306μs)
      • composing: site.headers.header (79μs)
      • Illuminate\Cache\Events\CacheHit (1.83ms)
      • creating: components.footer (106μs)
      • composing: components.footer (114μs)
      • Illuminate\Cache\Events\CacheHit (1.08ms)
      • Illuminate\Cache\Events\CacheHit (307μs)
      • Illuminate\Cache\Events\CacheHit (262μs)
      • Illuminate\Cache\Events\CacheHit (210μs)
      • Illuminate\Cache\Events\CacheHit (188μs)
      • Illuminate\Cache\Events\CacheHit (169μs)
      • Illuminate\Cache\Events\CacheHit (175μs)
      • Illuminate\Cache\Events\CacheHit (297μs)
      • Illuminate\Cache\Events\CacheHit (319μs)
      • Illuminate\Cache\Events\CacheHit (199μs)
      • Illuminate\Cache\Events\CacheHit (200μs)
      • Illuminate\Cache\Events\CacheHit (186μs)
      • Illuminate\Cache\Events\CacheHit (188μs)
      • Illuminate\Cache\Events\CacheHit (173μs)
      • creating: components.forms.contact-us (274μs)
      • composing: components.forms.contact-us (182μs)
      • creating: components.forms.get-started (218μs)
      • composing: components.forms.get-started (123μs)
      • creating: components.forms.free-tool-download (140μs)
      • composing: components.forms.free-tool-download (61μs)
      • creating: components.forms.claim-free-worksheet (126μs)
      • composing: components.forms.claim-free-worksheet (65μs)
      • creating: components.forms.tutor-subscription-waitlist (119μs)
      • composing: components.forms.tutor-subscription-waitlist (65μs)
      • creating: components.forms.tutor-subscription-join (121μs)
      • composing: components.forms.tutor-subscription-join (65μs)
      • creating: components.forms.tutor-support (119μs)
      • composing: components.forms.tutor-support (66μs)
      • 321 x Illuminate\Cache\Events\CacheHit (19.31%)
        84.73ms
        10 x Illuminate\Database\Events\StatementPrepared (5.66%)
        24.82ms
        10 x Illuminate\Database\Events\QueryExecuted (4.05%)
        17.79ms
        1 x Illuminate\Foundation\Events\LocaleUpdated (1.33%)
        5.82ms
        1 x eloquent.booting: App\Models\Subject (1.2%)
        5.28ms
        2 x eloquent.retrieved: App\Models\Subject (0.39%)
        1.69ms
        1 x Illuminate\Cache\Events\KeyWritten (0.33%)
        1.46ms
        1 x Illuminate\Database\Events\ConnectionEstablished (0.32%)
        1.38ms
        1 x Illuminate\Routing\Events\Routing (0.31%)
        1.37ms
        1 x Illuminate\Routing\Events\RouteMatched (0.13%)
        559μs
        1 x eloquent.booting: App\Models\SubjectCat (0.11%)
        491μs
        1 x creating: site.layouts.app (0.1%)
        437μs
        1 x creating: components.canonical (0.1%)
        420μs
        1 x creating: homework.show (0.09%)
        396μs
        1 x creating: components.breadcrumbs (0.08%)
        370μs
        1 x Illuminate\Cache\Events\CacheMissed (0.08%)
        339μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (0.08%)
        333μs
        1 x creating: site.headers.header (0.07%)
        306μs
        1 x creating: components.forms.contact-us (0.06%)
        274μs
        7 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (0.06%)
        262μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (0.06%)
        252μs
        1 x creating: components.forms.get-started (0.05%)
        218μs
        2 x eloquent.retrieved: App\Models\SubjectCat (0.05%)
        211μs
        1 x creating: components.open-graph (0.05%)
        203μs
        5 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.05%)
        201μs
        1 x composing: homework.show (0.04%)
        197μs
        1 x composing: components.breadcrumbs (0.04%)
        188μs
        1 x composing: components.forms.contact-us (0.04%)
        182μs
        1 x creating: components.forms.free-tool-download (0.03%)
        140μs
        1 x creating: components.forms.claim-free-worksheet (0.03%)
        126μs
        1 x composing: components.forms.get-started (0.03%)
        123μs
        1 x creating: components.forms.tutor-subscription-join (0.03%)
        121μs
        1 x creating: components.forms.tutor-subscription-waitlist (0.03%)
        119μs
        1 x creating: components.forms.tutor-support (0.03%)
        119μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.03%)
        117μs
        1 x composing: components.footer (0.03%)
        114μs
        1 x composing: components.canonical (0.02%)
        107μs
        1 x creating: components.footer (0.02%)
        106μs
        1 x composing: site.headers.header (0.02%)
        79μs
        1 x composing: components.open-graph (0.02%)
        78μs
        1 x composing: components.forms.tutor-support (0.02%)
        66μs
        1 x composing: components.forms.claim-free-worksheet (0.01%)
        65μs
        1 x composing: components.forms.tutor-subscription-waitlist (0.01%)
        65μs
        1 x composing: components.forms.tutor-subscription-join (0.01%)
        65μs
        1 x eloquent.booted: App\Models\Subject (0.01%)
        64μs
        1 x composing: components.forms.free-tool-download (0.01%)
        61μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.01%)
        47μs
        1 x eloquent.booted: App\Models\SubjectCat (0.01%)
        40μs
        1 x composing: site.layouts.app (0%)
        21μs
      14 templates were rendered
      • 1x homework.showshow.blade.phpblade
      • 1x components.breadcrumbsbreadcrumbs.blade.phpblade
      • 1x site.layouts.appapp.blade.phpblade
      • 1x components.canonicalcanonical.blade.phpblade
      • 1x components.open-graphopen-graph.blade.phpblade
      • 1x site.headers.headerheader.blade.phpblade
      • 1x components.footerfooter.blade.phpblade
      • 1x components.forms.contact-uscontact-us.blade.phpblade
      • 1x components.forms.get-startedget-started.blade.phpblade
      • 1x components.forms.free-tool-downloadfree-tool-download.blade.phpblade
      • 1x components.forms.claim-free-worksheetclaim-free-worksheet.blade.phpblade
      • 1x components.forms.tutor-subscription-waitlisttutor-subscription-waitlist.blade.phpblade
      • 1x components.forms.tutor-subscription-jointutor-subscription-join.blade.phpblade
      • 1x components.forms.tutor-supporttutor-support.blade.phpblade
      uri
      GET college-homework-library/{category}/{subject}/{id}
      middleware
      web, utm.parameters
      controller
      App\Http\Controllers\HomeworkLibraryController@show
      namespace
      where
      as
      homework.show
      file
      app/Http/Controllers/HomeworkLibraryController.php:79-176
      10 statements were executed, 4 of which were duplicates, 6 unique. Show only duplicated31.62ms
      • Connection Establishedtwenty4_siteHomeworkLibraryController.php#91
        Backtrace
        • 13. app/Http/Controllers/HomeworkLibraryController.php:91
        • 14. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 15. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 16. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary` where `status` = 'published' and `price` > 0 and `solutionslibrary`.`id` = '27276' limit 1
        15.19mstwenty4_siteHomeworkLibraryController.php#97
        Bindings
        • 0: published
        • 1: 0
        • 2: 27276
        Backtrace
        • 16. app/Http/Controllers/HomeworkLibraryController.php:97
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 18. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 19. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 20. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subjects` where `subjects`.`id` in (15)
        890μstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` in (27276)
        940μstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subject_cats` where `subject_cats`.`id` = 3 limit 1
        1.2mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 3
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 26. app/Http/Controllers/HomeworkLibraryController.php:105
        • 27. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 28. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 29. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
      • select * from `solutionslibrary` where `id` <> 27276 and `subject` = 15 and `status` = 'published' and `price` > 0 order by RAND() limit 6
        7.56mstwenty4_siteHomeworkLibraryRepository.php#30
        Bindings
        • 0: 27276
        • 1: 15
        • 2: published
        • 3: 0
        Backtrace
        • 14. app/Repositories/HomeworkLibraryRepository.php:30
        • 15. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 16. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 18. app/Repositories/HomeworkLibraryRepository.php:39
        • 19. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `subjects` where `subjects`.`id` in (15)
        1.1mstwenty4_siteHomeworkLibraryRepository.php#30
        Backtrace
        • 19. app/Repositories/HomeworkLibraryRepository.php:30
        • 20. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 21. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 23. app/Repositories/HomeworkLibraryRepository.php:39
        • 24. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 27276 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'question' order by `order` asc, `id` asc
        1.11mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 27276
        • 1: question
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:58
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 27276 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'teaser' order by `order` asc, `id` asc
        1.4mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 27276
        • 1: teaser
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:69
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 27276 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'solution' order by `order` asc, `id` asc
        1.07mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 27276
        • 1: solution
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:80
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `subject_cats` where `subject_cats`.`id` = 3 limit 1
        1.16mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 3
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 32. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 33. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 34. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69
        • 35. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
      App\Models\HomeworkLibrary\HomeworkLibrary
      7HomeworkLibrary.php
      App\Models\HomeworkLibrary\HomeworkLibraryFile
      5HomeworkLibraryFile.php
      App\Models\Subject
      2Subject.php
      App\Models\SubjectCat
      2SubjectCat.php
          _token
          RrAFttC9LRL8yIkfgeHgCeNSz9n64pjTSivVzLwl
          utm_source
          direct
          redirectUrl
          /college-homework-library/Computer-Science/Discrete-Math/27276
          _previous
          array:1 [ "url" => "https://staging.dev.24houranswers.com/college-homework-library/Computer-Scienc...
          _flash
          array:2 [ "old" => [] "new" => [] ]
          PHPDEBUGBAR_STACK_DATA
          []
          path_info
          /college-homework-library/Computer-Science/Discrete-Math/27276
          status_code
          200
          
          status_text
          OK
          format
          html
          content_type
          text/html; charset=UTF-8
          request_query
          []
          
          request_request
          []
          
          request_headers
          0 of 0
          array:21 [ "priority" => array:1 [ 0 => "u=0, i" ] "accept-encoding" => array:1 [ 0 => "gzip, deflate, br, zstd" ] "sec-fetch-dest" => array:1 [ 0 => "document" ] "sec-fetch-user" => array:1 [ 0 => "?1" ] "sec-fetch-mode" => array:1 [ 0 => "navigate" ] "sec-fetch-site" => array:1 [ 0 => "none" ] "accept" => array:1 [ 0 => "text/html,application/xhtml+xml,application/xml;q=0.9,image/avif,image/webp,image/apng,*/*;q=0.8,application/signed-exchange;v=b3;q=0.7" ] "user-agent" => array:1 [ 0 => "Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)" ] "upgrade-insecure-requests" => array:1 [ 0 => "1" ] "sec-ch-ua-platform" => array:1 [ 0 => ""Windows"" ] "sec-ch-ua-mobile" => array:1 [ 0 => "?0" ] "sec-ch-ua" => array:1 [ 0 => ""HeadlessChrome";v="129", "Not=A?Brand";v="8", "Chromium";v="129"" ] "cache-control" => array:1 [ 0 => "no-cache" ] "pragma" => array:1 [ 0 => "no-cache" ] "x-amzn-trace-id" => array:1 [ 0 => "Root=1-680a86d7-1c650899253e44e9232b5cb5" ] "host" => array:1 [ 0 => "staging.dev.24houranswers.com" ] "x-forwarded-port" => array:1 [ 0 => "443" ] "x-forwarded-proto" => array:1 [ 0 => "https" ] "x-forwarded-for" => array:1 [ 0 => "18.116.100.138" ] "content-length" => array:1 [ 0 => "" ] "content-type" => array:1 [ 0 => "" ] ]
          request_cookies
          []
          
          response_headers
          0 of 0
          array:5 [ "content-type" => array:1 [ 0 => "text/html; charset=UTF-8" ] "cache-control" => array:1 [ 0 => "no-cache, private" ] "date" => array:1 [ 0 => "Thu, 24 Apr 2025 18:45:43 GMT" ] "set-cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6IlpPTjRRS0lXKytpa3RINGh5NitNWWc9PSIsInZhbHVlIjoiVzAxcTFWV1hMNFBidDlYU1hjQi9iVytGTTNCaVNBSVFpZVA2WmppVUlLcG1SK2pXRWhwVDVnT29YRlA0MnEwQU84SEdyT1ZaRTRDZ3l1NjNieHE5K0Y3RXdHTWp6SkE4MjhFZnVwWkNRUGt2QzV0ekg5Tnk2VU4rT1F4TjJ2aGgiLCJtYWMiOiIwYzM0MmFjMTgzNjI5ZGZmODJiZTU2ZTFiNTgzMTlmZGJjMTZkMWEwNGRjZGZkMzMwNTFmZGI4ODM4YzY4M2JkIiwidGFnIjoiIn0%3D; expires=Thu, 24 Apr 2025 20:45:43 GMT; Max-Age=7200; path=/; domain=.24houranswers.com; samesite=laxXSRF-TOKEN=eyJpdiI6IlpPTjRRS0lXKytpa3RINGh5NitNWWc9PSIsInZhbHVlIjoiVzAxcTFWV1hMNFBidDlYU1hjQi9iVytGTTNCaVNBSVFpZVA2WmppVUlLcG1SK2pXRWhwVDVnT29YRlA0MnEwQU84SEdyT" 1 => "24houranswers_session=eyJpdiI6IktPamdaSFZoYW00ZUpsV1ZvWjRrVWc9PSIsInZhbHVlIjoicWY2YnA0R2ltYVRXaXRSeCt4UXlQUVVpaHN3Vmh0R3NwNTdwNWNBTGpUKytldlptWGE5S2RPSmdqNkNiWWhRc0wrRzc1MUxBWldGckxodUIzRThHaEtmUFU0Wm4xMWIzMmg4YTdQS2VrVkJBMElkSzV0eHB1L3FiT3M0WEg5ZjQiLCJtYWMiOiI1NGI0ZmM4YjEzNThjZmQ3NmE5YzZkNzFjNzAxN2U1NmZlZTY3NjMzNTYxOTc2MDQ5MmU5ZWQyNDZmOTJkMGVkIiwidGFnIjoiIn0%3D; expires=Thu, 24 Apr 2025 20:45:43 GMT; Max-Age=7200; path=/; domain=.24houranswers.com; httponly; samesite=lax24houranswers_session=eyJpdiI6IktPamdaSFZoYW00ZUpsV1ZvWjRrVWc9PSIsInZhbHVlIjoicWY2YnA0R2ltYVRXaXRSeCt4UXlQUVVpaHN3Vmh0R3NwNTdwNWNBTGpUKytldlptWGE5S2RPSmdqNkNiWW" ] "Set-Cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6IlpPTjRRS0lXKytpa3RINGh5NitNWWc9PSIsInZhbHVlIjoiVzAxcTFWV1hMNFBidDlYU1hjQi9iVytGTTNCaVNBSVFpZVA2WmppVUlLcG1SK2pXRWhwVDVnT29YRlA0MnEwQU84SEdyT1ZaRTRDZ3l1NjNieHE5K0Y3RXdHTWp6SkE4MjhFZnVwWkNRUGt2QzV0ekg5Tnk2VU4rT1F4TjJ2aGgiLCJtYWMiOiIwYzM0MmFjMTgzNjI5ZGZmODJiZTU2ZTFiNTgzMTlmZGJjMTZkMWEwNGRjZGZkMzMwNTFmZGI4ODM4YzY4M2JkIiwidGFnIjoiIn0%3D; expires=Thu, 24-Apr-2025 20:45:43 GMT; domain=.24houranswers.com; path=/XSRF-TOKEN=eyJpdiI6IlpPTjRRS0lXKytpa3RINGh5NitNWWc9PSIsInZhbHVlIjoiVzAxcTFWV1hMNFBidDlYU1hjQi9iVytGTTNCaVNBSVFpZVA2WmppVUlLcG1SK2pXRWhwVDVnT29YRlA0MnEwQU84SEdyT" 1 => "24houranswers_session=eyJpdiI6IktPamdaSFZoYW00ZUpsV1ZvWjRrVWc9PSIsInZhbHVlIjoicWY2YnA0R2ltYVRXaXRSeCt4UXlQUVVpaHN3Vmh0R3NwNTdwNWNBTGpUKytldlptWGE5S2RPSmdqNkNiWWhRc0wrRzc1MUxBWldGckxodUIzRThHaEtmUFU0Wm4xMWIzMmg4YTdQS2VrVkJBMElkSzV0eHB1L3FiT3M0WEg5ZjQiLCJtYWMiOiI1NGI0ZmM4YjEzNThjZmQ3NmE5YzZkNzFjNzAxN2U1NmZlZTY3NjMzNTYxOTc2MDQ5MmU5ZWQyNDZmOTJkMGVkIiwidGFnIjoiIn0%3D; expires=Thu, 24-Apr-2025 20:45:43 GMT; domain=.24houranswers.com; path=/; httponly24houranswers_session=eyJpdiI6IktPamdaSFZoYW00ZUpsV1ZvWjRrVWc9PSIsInZhbHVlIjoicWY2YnA0R2ltYVRXaXRSeCt4UXlQUVVpaHN3Vmh0R3NwNTdwNWNBTGpUKytldlptWGE5S2RPSmdqNkNiWW" ] ]
          session_attributes
          0 of 0
          array:6 [ "_token" => "RrAFttC9LRL8yIkfgeHgCeNSz9n64pjTSivVzLwl" "utm_source" => "direct" "redirectUrl" => "/college-homework-library/Computer-Science/Discrete-Math/27276" "_previous" => array:1 [ "url" => "https://staging.dev.24houranswers.com/college-homework-library/Computer-Science/Discrete-Math/27276" ] "_flash" => array:2 [ "old" => [] "new" => [] ] "PHPDEBUGBAR_STACK_DATA" => [] ]