QuestionFind the inverse function $f^{-1}$ for $f(x)=\sqrt[3]{x-2}+8$ .

Studdy Solution

STEP 1

Assumptions1. The function is $f(x)=\sqrt[3]{x-}+8$ . We are asked to find the inverse of this function, denoted as $f^{-1}(x)$

STEP 2

To find the inverse of a function, we first replace $f(x)$ with $y$ .
$y = \sqrt[]{x-2}+8$

STEP 3

Next, we swap $x$ and $y$ to find the inverse.
$x = \sqrt[3]{y-2}+8$

STEP 4

Now, we need to solve this equation for $y$ to find $f^{-1}(x)$ .
First, isolate the cube root term by subtracting8 from both sides.
$x -8 = \sqrt[3]{y-2}$

STEP 5

Next, we eliminate the cube root by cubing both sides of the equation.
$(x -8)^3 = (y-2)$

STEP 6

Finally, we solve for $y$ by adding2 to both sides of the equation.
$(x -8)^3 +2 = y$ So, the inverse of the function $f(x)$ is $f^{-1}(x) = (x -8)^3 +2$ .

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