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Chapter 1 | Functions and Graphs
Finding a Function’s Inverse We can now consider one-to-one functions and show how to find their inverses. Recall that a function maps elements in the domain of f to elements in the range of f . The inverse function maps each element from the range of f back to its corresponding element from the domain of f . Therefore, to find the inverse function of a one-to-one function f , given any y in the range of f , we need to determine which x in the domain of f satisfies f ( x ) = y . Since f is one-to-one, there is exactly one such value x . We can find that value x by solving the equation f ( x ) = y for x . Doing so, we are able to write x as a function of y where the domain of this function is the range of f and the range of this new function is the domain of f . Consequently, this function is the inverse of f , and we write x = f −1 ( y ). Since we typically use the variable x to denote the independent variable and y to denote the dependent variable, we often interchange the roles of x and y , andwrite y = f −1 ( x ). Representing the inverse function in this way is also helpful later when we graph a function f and its inverse f −1 on the same axes.
Problem-Solving Strategy: Finding an Inverse Function 1. Solve the equation y = f ( x ) for x . 2. Interchange the variables x and y and write y = f −1 ( x ).
Example 1.29 Finding an Inverse Function
Find the inverse for the function f ( x ) =3 x −4. State the domain and range of the inverse function. Verify that f −1 ( f ( x )) = x .
Solution Follow the steps outlined in the strategy. Step 1. If y =3 x −4, then 3 x = y +4 and x = 1 3
y + 4
3 .
and let y = f −1 ( x ).
Step 2. Rewrite as y = 1 3
x + 4 3
Therefore, f −1 ( x ) = 1 3
x + 4
3 . Since the domain of f is (−∞, ∞), the range of f −1 is (−∞, ∞). Since the range of f is (−∞, ∞), the domain of f −1 is (−∞, ∞). You can verify that f −1 ( f ( x )) = x by writing f −1 ( f ( x )) = f −1 (3 x −4) = 1 3 (3 x −4)+ 4 3 = x − 4 3 + 4 3 = x . Note that for f −1 ( x ) to be the inverse of f ( x ), both f −1 ( f ( x )) = x and f ( f −1 ( x )) = x for all x in the domain of the inside function.
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