1. Sec. 2.7 Inverse Functions

2. Inverse of a function
Found by interchanging the coordinates in each ordered pair
Denoted by f -1

3. Inverse functions f(x) : {(1,5), (2,6), (3,7), (4,8)}
Note: the domain of a f is equal to the range of f -1 and vice versa

4. Inverse functions undo each other
Inverse functions undo each other. When you form the composite of inverse functions you get the identity function

5. Write the inverse function of
f(x) : {(3,5), (-2,1), (-5,3), (-4,-1)}
f -1(x): {(5,3), (1,-2), (3,-5), (-1,-4)}

6. Graph the f(x) and f -1(x) we just found
Note: the graph f -1is a reflection of the graph of f over the line y=x

7. To determine if 2 functions are inverses
Graph – if they are a reflection over line y=x then they are inverses
Algebraically – if (f ◦ f -1)(x)=(f -1 ◦f)(x)=x
f(f -1(x))= f -1(f(x))=x
then they are inverses

8. Example
f(x)=3-4x g(x)=(3-x)/4

9. Ex. 3 Verify Inverses
f(x)=2x3-1
g(x)= 3√(x+1)/2

10. To find the inverse of a function
Replace f(x) with y
Interchange x & y
Does the new equation represent y as a function of x?
If NO - f does not have an inverse
If YES solve for y
Replace y with f -1(x)
Verify that f & f -1are inverses by showing that the domain of f is = to the range of f -1& range of f = domain of f -1

11. Ex. 5 Find the inverse of f(x)=(5-3x)/2
Step 1 y=(5-3x)/2
Step 2 x=(5-3y)/2
Step 3 yes, y is a function of x (solve for y)
Step 4 f -1(x)=-2/3 x +5/3
Step 5 verify

12. Ex. Find f -1(x) for f(x)=5x-7
y=5x-7
x=5y-7
x+7=5y
5 5
y=(x+7)/5
Yes y is a function of x
f -1(x)=(x+7)/5

13. Horizontal Line Test
A function has an inverse if and only if no horizontal line intersects the graph of f at more than 1 point

14. Homework p. 243 1-4, 17-20 (just algebraically, 29-32, 33-37 odd, 39-59odd (don’t graph)