Inverse Functions. Objectives. Students will be able to find inverse functions and verify that two functions are inverse functions of each other. Students will be able to use the graph of a functions to determine whether functions have inverse functions.
Let f and g be two functions such that
f (g(x)) = x for every x in the domain of g
g(f (x)) = x for every x in the domain of f.
The function g is the inverse of the functionf, 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.Definition of the Inverse Function
f (x) = 3x and g(x) = x/3.
To show that f and g are inverses of each other, we must show that f (g(x)) = x and g( f (x)) = x. We begin with f (g(x)).
f (x) = 3x
f (g(x)) = 3g(x) = 3(x/3) = x.
Next, we find g(f (x)).
g(x) = x/3
g(f (x)) = f (x)/3 = 3x/3 = x.
Notice how f -1 undoes the change produced by f.
This graph does not pass the horizontal line test, so f(x) = x2+3x-1 does not have an inverse function.Example
The equation for the inverse of a function f can be found as follows:
Step 1 Replace f (x) by y.
y = 6x + 3
Step 2 Interchange x and y.
x = 6y + 3
This is the inverse function.
Step 3 Solve for y.
x - 3 = 6y
Subtract 3 from both sides.
x - 3 = y
Divide both sides by 6.
Step 4 Replace y by f -1(x).
x - 3
f -1(x) =
Rename the function f -1(x).
Find the inverse of f (x) = 6x + 3.