Let $$ f(x)=\sqrt[5]{5 x-3} $$. Determine $$ f^{-1}(x) $$ $$ f^{-1}(x)=\square $$

Question

UpStudy AI · Accepted Answer

Let $$ f(x)=\sqrt[5]{5x-3} $$. We want to find $$ f^{-1}(x) $$, which satisfies
$$
f\big(f^{-1}(x)\big)=x.
$$

**Step 1.** Write the function in terms of $$ y $$:
$$
y = \sqrt[5]{5x-3}.
$$

**Step 2.** Replace $$ x $$ with $$ y $$ and $$ y $$ with $$ x $$ to set up the equation for the inverse function:
$$
x = \sqrt[5]{5y-3}.
$$

**Step 3.** Eliminate the fifth root by raising both sides of the equation to the power of 5:
$$
x^5 = 5y - 3.
$$

**Step 4.** Solve for $$ y $$:
$$
5y = x^5 + 3 \quad \Longrightarrow \quad y = \frac{x^5+3}{5}.
$$

Thus, the inverse function is:
$$
f^{-1}(x) = \frac{x^5+3}{5}.
$$
$$ f^{-1}(x) = \frac{x^5 + 3}{5} $$

Let \( f(x)=\sqrt[5]{5 x-3} \). Determine \( f^{-1}(x) \) \( f^{-1}(x)=\square \)