L 2.6.2 Test (CST): Functions and Relations
Question 5 of 25
$f(x)=\sqrt[3]{x+13}$. Find the inverse of $f(x)$
A. $f^{-1}(x)=(x-13)^{3}$
B. $f^{-1}(x)=x^{3}+13$
c. $f^{-1}(x)=x^{3}-13$
D. $f^{-1}(x)=(x+13)^{3}$

Step 1 ：The function is given as \(f(x) = \sqrt[3]{x+13}\).

Step 2 ：To find the inverse of a function, we swap the x and y (or f(x)) values and solve for y. So, we replace f(x) with y and x with f(x) in the equation, giving us \(x = \sqrt[3]{y+13}\).

Step 3 ：We then solve this equation for y to find the inverse function. To do this, we cube both sides of the equation to get rid of the cube root on the right side, resulting in \(x^{3} = y + 13\).

Step 4 ：Finally, we subtract 13 from both sides to solve for y, giving us \(y = x^{3} - 13\).

Step 5 ：So, the inverse function of \(f(x)\) is \(f^{-1}(x) = x^{3} - 13\).

Step 6 ：\(\boxed{f^{-1}(x) = x^{3} - 13}\) is the final answer.