1. 49 – The Inverse Function No Calculator
Logarithmic Investigations
49 – The Inverse Function No Calculator

2. The Existence of the Inverse of f(x)NO
DO ALL functions have an inverse function?
The Existence of the Inverse of f(x)
IF for every x there is at most one y (function in terms of x)
AND
IF for every y there is at most one x (function in terms of y)
the function is called one-to-one.
A function which is one-to-one will have an inverse.

3. All topics in calculus must be studied from three different perspectives:
Graphically (graphs)
Tabularly (tables)
Analytically (equations)
Let’s look at the existence of the inverse from these three perspectives.
(We’ll find the inverse itself later.)

4. Existence of Inverse - GRAPHICALLY
Inverse Exists
Function in terms of x
AND
Function in terms of y
No Inverse Exists
(1, 1), (-1, 1)
No Inverse Exists
(1, 0), (0, 0)

5. Does the inverse exist? Does the inverse exist? Inverse Exists
one-to-one
No Inverse Exists
(3, 0.9), (7, 0.9)

6. Does the inverse exist? Does the inverse exist? Inverse Exists
one-to-one
Inverse Exists
one-to-one

7. Does the inverse exist? Does the inverse exist? No Inverse Exists
(-3, 2), (0, 2)
Inverse Exists
one-to-one

8. Existence of Inverse - TABULARLY
January
February
March
July
Winter
Spring
Summer
No Inverse Exists
(1, 2), (2, 2)
No Inverse Exists
(Jan, Winter), (Feb, Winter)
Ford
Bush
Carter
Clinton
President
Vice-President
Inverse Exists
one-to-one
No Inverse Exists
(Ford, President),
(Ford, Vice-President)
No Inverse Exists
(6, 5), (6, 3)

9. Existence of Inverse - ANALYTICALLY
No Inverse Exists
(-4, -12), (-2, -12)
No Inverse Exists
(4, 2), (4, -2)
Inverse Exists
one-to-one
No Inverse Exists
(0, 2), (0, -2)
No Inverse Exists
(4, 0), (-4, 0)

10. FINDING the inverse function
1. Determine if inverse function exists (one-to-one).
If yes, proceed to #2.
2. Switch the x and the y.
3. (Analytically) Solve for the ‘new’ y.

11. Finding the Inverse Function - GRAPHICALLY
See PowerPoint Notes

12. Find the inverse of f(x).

13. Finding the Inverse Function - TABULARLY
No Inverse Exists
(5, 4), (7, 4)

14. Finding the Inverse ANALYTICALLY

15. YES Composition of Inverse Functions
If two one-to-one functions are inverses of each other:
CHECK: f(x) and g(x) are both one-to-one
YES

16. CHECK: f(x) is NOT one-to-one

17. CHECK: f(x) and g(x) are both one-to-one
YES