Math Snap

STEP 1

1. We are given the function $f(x) = \sqrt[5]{x} - 2$.
2. We need to find the inverse function $f^{-1}(x)$.

STEP 2

1. Set $y = f(x)$ and express it in terms of $x$.
2. Solve for $x$ in terms of $y$.
3. Swap $x$ and $y$ to find $f^{-1}(x)$.
4. Verify the inverse function.

STEP 3

Set $y = f(x)$:
$$y = \sqrt[5]{x} - 2$$

STEP 4

Solve for $x$ in terms of $y$:
First, isolate the radical term:
$$y + 2 = \sqrt[5]{x}$$ Next, eliminate the fifth root by raising both sides to the power of 5:
$$(y + 2)^5 = x$$

STEP 5

Swap $x$ and $y$ to find the inverse function:
$$f^{-1}(x) = (x + 2)^5$$

Verify the inverse function by checking if $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.
First, check $f(f^{-1}(x))$:
$$f(f^{-1}(x)) = f((x + 2)^5) = \sqrt[5]{(x + 2)^5} - 2 = x + 2 - 2 = x$$ Next, check $f^{-1}(f(x))$:
$$f^{-1}(f(x)) = f^{-1}(\sqrt[5]{x} - 2) = ((\sqrt[5]{x} - 2) + 2)^5 = (\sqrt[5]{x})^5 = x$$ Both conditions are satisfied, confirming the inverse.
The inverse function is:
$$f^{-1}(x) = (x + 2)^5$$