4 5 inverse functions
- 1. 4-5 INVERSE FUNCTIONS Objectives: 1. Find the inverse of a function, if it exists.
- 2. INVERSE EXAMPLE Conversion formulas come in pairs, for example: These formulas “undo” each other, so they are inverses.
- 3. DEFINITION OF INVERSES Two functions f and g are inverses if: f(g(x)) = x and g(f(x)) = x To check if two functions are inverses, perform both compositions and make sure both equal x.
- 4. EXAMPLE 1 If and show that f and g are inverses.
- 6. INVERSE NOTATION The inverse of f is written f -1 f -1(x) Note: is the value of f -1 at x is not
- 7. FINDING INVERSES graph of f -1 is the reflection of f over the line y = x Can be found by switching x and y in the ordered pairs. Find equation of f -1 by switching x and y in the equation and solving for y. The
- 8. EXAMPLE 2 f(x) = 4 – x2 for x 0. Sketch the graph of f and f -1 (x) Find a rule for f -1 (x) Let
- 9. YOU TRY! g(x) = (x – 4)2 – 1 for x 4. Sketch the graph of g and g -1 (x) Find a rule for g -1 (x) Let
- 10. EXAMPLE 3 Suppose a function f has an inverse. If f(2) = 3, find: f -1 (3) f(f -1(3)) f -1(f(2))
- 11. YOU TRY! Suppose a function g has an inverse. If g(5) = 1, find: g -1 (1) g -1(g(5)) g(g -1(1))
- 12. DO ALL FUNCTIONS HAVE INVERSES? the graph of y = x2 over the line y = x. Is the result a function? Reflect
- 13. ONE-TO-ONE Only functions that are one-to-one have inverses. One-to-one means each x value has exactly one y value and each y has exactly one x Can check using horizontal line test.
- 14. EXAMPLE 4 Which functions are one-to-one? Which have inverses?
- 15. YOU TRY! Is h(x) one-to-one? Does it have an inverse?
- 16. EXAMPLE 5 State whether each function has an inverse. If yes, find f -1 (x) and show f(f -1(x)) = f -1(f(x)) = x
- 17. YOU TRY! Does have an inverse? If so, find f -1 (x).