1. Section 1.9 Inverse Functions

2. What you should learn
How to find inverse functions informally and verify that two functions are inverse functions of each other
How to use graphs of functions to determine whether functions have inverse functions
How to use the horizontal line test to determine if functions are one-to-one
How to find inverse functions algebraically

3. Consider the function, f(x) that doubles x and then subtracts 4.-4 2 4
Now swap x and y to create a new function g(x) x
g(x) -4 2 4
This function needs to add 4 first, then divide by 2

4. Inverse Notation f -1(x) is read as the f-inverse
Notice that an inverse of a function does the opposite thing in the opposite order of the original function.

5. Composition of Inverses
What is the name of this function?

6. Definition of Inverse Function
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.
Under these conditions the function g is the inverse function of the function f.

7. Verifying Inverse Functions
To verify that two functions f and g are inverse functions, form the composition.
If the composition yields the identity function then the two functions are inverses of each other.
Therefore g(x) is not the inverse of f(x).

8. Graphs of Inverse Functionsx
f(x) -4 2 4 x
g(x) -4 2 4

9. Vertical Line Test? x
f(x) -2 -4 2

10. Vertical Line Test? x f(x) -2 -4 2 x f-1(x) -2 -4 2-4 2 x
f-1(x) -2 -4 2
The inverse relation does not pass the vertical line test.
Not a function.

11. Horizontal Line Test
A function f has an inverse function if and only if no horizontal line intersects the graph of f at more than one point.
The inverse relation will not be a function.

12. One-to-One
A function f is one-to-one if each value of the dependent variable corresponds to exactly one value of the independent variable.
A function f has in inverse function if and only if f is one-to-one.

13. One-to-One?

14. One-to-One?

15. One-to-One?

16. One-to-One?

17. One-to-One? PASS FAIL Horizontal Line Test Vertical Line Test NOT

18. One-to-One? One-to-One PASS Pass Horizontal Line Test
Vertical Line Test
One-to-One

19. Finding Inverse Functions Algebraically
Use the horizontal line test to decide whether f has an inverse function.
Replace f(x) with y.
Swap x and y, and solve for y.
Replace y with f -1(x).
Verify that the range of one is the domain of the other.

20. Find f -1(x) Replace f(x) with y Swap x with y Solve for y
Replace y with f -1 (x)