Understanding Composite and Inverse Functions

Learn about composite functions, inverse functions, and how to find their compositions and inverses through examples and step-by-step explanations. Explore the concept of forming composite functions, verifying inverse functions, and finding the inverse of a function using interchange and solving methods. Discover the horizontal line test and one-to-one functions to deepen your understanding of function operations.

• Functions
• Composite
• Inverse
• Composition
• Verification

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1. Chapter 9 Section 2 Composite and Inverse Functions Page 676

2. Warm up 1) If f(x) = 3x 2 find f(4). 2) If g(x) = 2?2+ 7? 6 , find g(3)

3. Composite Functions Rewrite two functions as one. Notation: composite function ( ) f g x ( ) the composition of the function f with g. This is read, f of g of x x is in the domain of g g(x) is in the domain of f

4. Example f(x) = x 300, g(x) = 0.85x find Solution: Since is the same as f(g(x)) Replace g(x) with 0.85x f(0.85x) So f(0.85x) = 0.85x 300 Then write the answer: =0.85x - 300

5. Form a Composite Function x2+6 Given: f(x) = 3x 4 and g(x) = Find and g(f(x))

6. One More Find (? ?) (x) if f(x) = 2x and g(x) = x + 7

7. Inverse Functions f-1 Notation: Definition: Let f and g be two functions such that f(g(x)) = x for every x in the domain of g g(f(x)) = x for every x in the domain of f Function g is the inverse of the function f read f inverse . f-1

8. Verify Inverse Functions x 5 f(x) = 5x and g(x) = To verify that f(x) and g(x) are inverses, show that f(g(x) = x and g(f(x)) = x

9. Find the Inverse of a Function 1) Replace f(x) with y in the equation. 2) Interchange x and y 3) Solve for y If the function does not have an inverse, stop. 4) If f has an inverse, then replace y with f-1x ( )

10. Find the inverse a) f(x) = 2x + 3 b) f(x) = ?3 1

11. Horizontal Line Test and One-toOne Functions f-1 A function f has an inverse that is a function, , if there is no horizontal line that interests the graph of the function f at more than one point. If the function passes the horizontal line test, the function is call a one- to-one function.

12. Which graph passes the Horizontal Line Test? a, b, c, d ?

13. Which graphs represent functions that have inverse function? Explain how you know.

14. Summary Composite functions. Inverse functions. Horizontal line test.