Table of contents
- 0. Review of Algebra4h 16m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 6m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 19m
- 10. Combinatorics & Probability1h 45m
3. Functions
Function Composition
3:15 minutes
Problem 1
Textbook Question
Textbook QuestionIn Exercises 1-10, find f(g(x)) and g (f(x)) and determine whether each pair of functions ƒ and g are inverses of each other. f(x) = 4x and g(x) = x/4
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Composition
Function composition involves combining two functions, where the output of one function becomes the input of another. For functions f and g, the composition f(g(x)) means applying g first and then f to the result. This process is essential for evaluating how two functions interact and can reveal properties such as inverses.
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Inverse Functions
Inverse functions are pairs of functions that 'undo' each other. If f(x) takes an input x and produces an output y, then g(y) should return the original input x. To determine if two functions are inverses, one must check if f(g(x)) = x and g(f(x)) = x for all x in their domains.
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Linear Functions
Linear functions are polynomial functions of degree one, typically expressed in the form f(x) = mx + b, where m is the slope and b is the y-intercept. In this case, f(x) = 4x and g(x) = x/4 are both linear functions, which simplifies the process of finding their compositions and checking for inverses due to their predictable behavior and straightforward algebraic manipulation.
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