Precalculus 1.7

1. Precalculus 1.7 INVERSE FUNCTIONS

2. DO THIS NOW! • You have a function described by the equation: f(x) = x + 4 • The domain of the function is: {0, 2, 5, 10} • YOUR TASK: write the set of ordered pairs that would represent this function • **use the equation to find f(x) for each number in the domain • Graph the points in the function.

3. Functions as Sets of Ordered Pairs • Recall that besides describing functions with equations or graphs, we can do so by listing the ordered pairs that make up the function

4. Inverse function • Notation: f-1 is the inverse of f • Definition: A function’s inverse brings output values back to their input values. f(x)=x+4 6 2 INPUT OUTPUT f-1(x)=x-4

5. Inverse Functions: 3 representations GRAPH ALGEBRA ORDERED PAIRS

6. Inverse Functions: Ordered Pairs • Original function, f: {(0, 4) (2, 6) (5, 9) (10, 14)} • Inverse function, f-1: {(4, 0) (6, 2) (9, 5) (14, 10)} • What is f(2)? • What is f-1(6)? • To find the inverse function, represented by ordered pairs, simply flip each ordered pair • If f contains (x, y), then f-1 contains (y, x).

7. Inverse Functions: Algebra • The equation of the inverse function should “undo” the equation of the original function. • Ex: If f(x) = x + 4, then f-1(x) = x – 4 • Ex: If g(x) = 4x, then g-1(x) = …?

8. Precise Definition Let f and g be two functions such that f(g(x)) = x for every x in the domain of g and g(f(x)) = x for every x in the domain of f. Then g is the inverse of the function f. g = f-1 The domain of f is the range of f-1. The range of f is the domain of f-1.

9. A logical point: If f is the inverse of g, then g is the inverse of f. Furthermore, g and f are inverses of each other.

10. Example • Which of these is the inverse of:

11. Inverse Function: GRAPHS • Graph these functions and their inverses on the same graph. • f(x) = x + 4 and g(x) = x – 4 • f(x) = 4x and g(x) = x/4 • f(x) = .5(x+3) and g(x) = 2x – 3 • f(x) = x2 and • {(0,1) (2,5) (3,6) (4,8)} and {(1,0) (5,2) (6,3) (8,4)}

12. Inverse Graphs • The graph of a f and f-1 are related in a special way. • If (x,y) is on f’s graph, (y,x) must be on f-1’s graph. • Therefore, the graph of f-1 is a reflection of the graph of f across the line y = x.

13. Practice • What is the inverse of: f(x) = 2x + 4 • What is the inverse of: f(x) = 1/x

14. Which functions have inverses? What is this function’s inverse? (0, 1) (2, 4) (3, 4)

15. Functions without inverses • If multiple input values have the same output value, then the function has no inverse • This is because given a repeated output, there would be no way to tell what the original input was • f: (0,1) (2,4) (3,4) • f-1: (1,0) (4,2) (4,3) This is NOT a function! Therefore we have no surefire way to undo f.

16. Functions without inverses: a list • f(x) = x2 (What was x if f(x)=4?) • f(x) = xn where n is even • f(x) = |x|

17. Functions without inverses: graph test If a function, f, has an inverse, f-1, then the inverse is also a function. Therefore f-1 must pass the vertical line test. In order for f-1 to pass the vertical line test, f must pass the HORIZONTAL LINE TEST. f-1 f

18. Will these functions have inverses? 1) 2) 3) 4) If a function both increases and decreases, can it have an inverse?

19. Finding the inverse of a function algebraically 1. 2. 3. 4. 5. • Use the horizontal line test to decide whether f has an inverse. • In the equation, replace f(x) with “y.” • Switch “x” and “y.” • Solve for y. • Replace y with “f-1(x).” • Check your work!

20. Practice: find the inverses of these functions 2) 1) 3) 4) 5) 6)