3.7-2 – Inverse Functions and Properties

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

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

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

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

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

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

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

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

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

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