1. 3.7-2 – Inverse Functions and Properties

2. We know how to graph the inverse of a function, but now we will look into expressing a new inverse function
Like before, let’s keep in mind the “switching x and y” theory

3. f-1(x)
The inverse of the function f(x), f-1(x), can be found by switching x and y, and resolving for y
Replace f(x), or whatever the function name is, with y
For example, for the function f(x) = x2, I would write the new equation x = y2, and solve for y

4. Horizontal Line Keep in mind the horizontal line test!
If it fails, then…

5. Example. Find the inverse for the function f(x) =

6. Example. Find the inverse of the function f(x) = (x+2)5 + 9

7. Properties of Inverses
For any inverse function, the following properties exist:
f(f-1(x)) = x, for all x in the domain for f-1
f-1(f(x)) = x, for all x in the domain for f
Tells us that if we evaluate the two functions, in either order, we get out x
Used as a test to confirm the inverse

8. Properties of Inverses
All graphs of functions who are inverses of each other are reflected over the line y = x
Confirmed by the previous slide
An inverse “undoes” the other function

9. Example. Show the following two functions are inverses of each other.
f(x) = (3x-1)/5 f-1(x) = (5x + 1)/3

10. Example. Show the following two functions are inverses of each other.
f(x) = f-1(x) = x3 - 1

11. Assignment
Pg. 283
31 – 57 odd, 63, 64