1. Inverse Functions

2. Function
If for every x there exists at most one y
One – to – One Function
If for every x there exists at most one y AND
for every y there exists at most one x
Function
Function
One – to – One
NOT One – to – One

3. Only one – to – one functions have inverses
GRAPHICALLY
One – to – One Function
Inverse

4. Function and Inverse have Symmetry about the line y = x
One – to – One Function
Inverse
Function and Inverse have
Symmetry about the line y = x

5. Find the inverse graph of the function below, if it exists.
Since not ONE – TO – ONE, no inverse function exists

6. Find the inverse graph of the function below, if it exists.
(4.5, 7)
(7, 4.5)
(0, 3)
(-4, 1)
(3, 0)
(1, -4)
(-7, -5)
(-5, -7)

7. Find the inverse graph of the function below, if it exists.

8. Finding inverse functions algebraically

9. Finding inverse functions algebraically
PCH ONLY – LOOK AT PROBLEM 68

10. Derivatives of Inverse Functions
If f is differentiable at every point on an interval, and f ’ is never
zero on the interval, then:
is differentiable at every point on the interior of the
Interval and its value at the point f(x) is:

11. Find the derivative of the inverse of f(x) = 5 – 4x evaluated
at c = ½

12. Find the derivative of the inverse of evaluated
at c = 4
Not One-to-One
However….One-to-One on [2, 6] which not only includes
c = 4, but it also eliminates f ‘ (x) = 0 issue as well
Alternative…….

13. Find the derivative of the inverse of evaluated
at c = 4

14. Find the derivative of the inverse of at c = 1