Select the correct answer.
Which function is the inverse of [tex]$f(x)=-x^3-9$[/tex]?
A. [tex]$f^{-1}(x)=\sqrt[3]{x+9}$[/tex]
B. [tex]$f^{-1}(x)=\sqrt[3]{-x-9}$[/tex]
C. [tex]$f^{-1}(x)=-\sqrt[3]{-x+9}$[/tex]
D. [tex]$f^{-1}(x)=-\sqrt[3]{x-9}$[/tex]
Answer (1)
Replace f ( x ) with y and get y = − x 3 − 9 .
Swap x and y to get x = − y 3 − 9 .
Solve for y : y = 3 − x − 9 .
The inverse function is f − 1 ( x ) = 3 − x − 9 , so the answer is B .
Explanation
Understanding the Problem We are given the function f ( x ) = − x 3 − 9 and we need to find its inverse, f − 1 ( x ) . The inverse function is found by swapping x and y and then solving for y .
Swapping x and y Let y = f ( x ) , so we have y = − x 3 − 9 . To find the inverse, we swap x and y to get x = − y 3 − 9 .
Isolating the y term Now we solve for y . First, add 9 to both sides: x + 9 = − y 3 .
Multiplying by -1 Next, multiply both sides by -1: − x − 9 = y 3 .
Taking the cube root Finally, take the cube root of both sides: y = 3 − x − 9 .
Finding the Inverse Function Therefore, the inverse function is f − 1 ( x ) = 3 − x − 9 . Comparing this to the given options, we see that it matches option B.
Examples
Imagine you're baking a cake and need to convert the oven temperature from Celsius to Fahrenheit. Finding the inverse of a function is like converting back from Fahrenheit to Celsius. In this case, if f ( x ) represents the Celsius to Fahrenheit conversion, then f − 1 ( x ) would represent the Fahrenheit to Celsius conversion. This concept is useful in many real-world scenarios where you need to reverse a process or calculation.
