Inverse Functions

Definition of the 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. The function g is the inverse of the function f, and denoted by f -1 (read “f-inverse”). Thus, f ( f -1(x)) = x and f -1( f (x)) = x. The domain of f is equal to the range of f -1, and vice versa.

Text Example Show that each function is the inverse of the other: f (x) = 5x and g(x) = x/5. Solution To show that f and g are inverses of each other, we must show that f (g(x)) = x and g( f (x)) = x. f (g(x)) = 5g(x) = 5(x/5) = x. g(f (x)) = f (x)/5 = 5x/5 = x. Notice how f -1 undoes the change produced by f.

Finding the Inverse of a Function The equation for the inverse of a function f can be found as follows: Replace f (x) by y in the equation for f (x). Interchange x and y. Solve for y. If this equation does not define y as a function of x, the function f does not have an inverse function and this procedure ends. If this equation does define y as a function of x, the function f has an inverse function. If f has an inverse function, replace y in step 3 with f -1(x). We can verify our result by showing that f ( f -1(x)) = x and f -1( f (x)) = x.

Text Example Find the inverse of f (x) = 7x – 5. Solution Step 1 Replace f (x) by y. y = 7x – 5 Step 2 Interchange x and y. x = 7y – 5 This is the inverse function. Step 3 Solve for y. x + 5 = 7y Add 5 to both sides. x + 5 = y 7 Divide both sides by 7. Step 4 Replace y by f -1(x). x + 5 7 f -1(x) = Rename the function f -1(x).

One-to-One Functions If f is a one-to-one function then its inverse will also be a function. A function is one-to-one if for different values of x there are different values of f(x). The easiest way to test for this property is to use the horizontal line test.

The Horizontal Line Test For Inverse Functions A function f has an inverse that is a function, f –1, if there is no horizontal line that intersects the graph of the function f at more than one point.

Example Does f(x) = x2+3x-1 have an inverse function?

Example Solution: This graph does not pass the horizontal line test, so f(x) = x2+3x-1 does not have an inverse function.

Example The graph an inverse function is the reflection of the original function about the line y = x. Graph the following functions and their inverse functions on the same axes: