Find the inverse of the given function. $$ f(x)=-4 x^{5}+2 $$ $$ f^{-1}(x)=\square $$

Question

UpStudy AI · Accepted Answer

We start with the function  
$$
f(x) = -4 x^5 + 2.
$$

Let  
$$
y = -4 x^5 + 2.
$$

**Step 1.** Subtract 2 from both sides:
$$
y - 2 = -4 x^5.
$$

**Step 2.** Multiply both sides by $$-1$$:
$$
2 - y = 4 x^5.
$$

**Step 3.** Divide both sides by 4:
$$
x^5 = \frac{2-y}{4}.
$$

**Step 4.** Take the fifth root of both sides (recall that for odd roots this is valid for all real numbers):
$$
x = \sqrt[5]{\frac{2-y}{4}}.
$$

Now, to write the inverse function, switch the roles of $$x$$ and $$y$$:
$$
y = \sqrt[5]{\frac{2-x}{4}}.
$$

Thus, the inverse function is:
$$
f^{-1}(x) = \sqrt[5]{\frac{2-x}{4}}.
$$
The inverse function is $$ f^{-1}(x) = \sqrt[5]{\frac{2 - x}{4}} $$.

Find the inverse of the given function. \( f(x)=-4 x^{5}+2 \) \( f^{-1}(x)=\square \)