QuestionFind the inverse function $f^{-1}$ of $f(x)=\left(x^{4}+4\right)^{-1}$ for $x \in (-\infty, 0]$ .

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is given by $f(x)=\left(x^{4}+4\right)^{-1}$ for $x \in (-\infty,0]$ . We need to find the inverse function $f^{-1}(x)$

STEP 2

To find the inverse of a function, we replace $f(x)$ with $y$ .
$y = \left(x^{4}+4\right)^{-1}$

STEP 3

Next, we interchange $x$ and $y$ to find the inverse.
$x = \left(y^{}+\right)^{-1}$

STEP 4

To solve for $y$ , we first take the reciprocal of both sides.
$\frac{1}{x} = y^{4}+4$

STEP 5

Subtract4 from both sides.
$\frac{1}{x} -4 = y^{4}$

STEP 6

Take the fourth root of both sides to solve for $y$ . $y = \sqrt[4]{\frac{1}{x} -4}$

STEP 7

However, since the original function $f(x)$ was defined for $x \leq0$ , the inverse function $f^{-1}(x)$ must be defined for $y \leq0$ . Therefore, we take the negative of the fourth root.
$y = -\sqrt[4]{\frac{1}{x} -4}$ So, the inverse function $f^{-1}(x)$ is given by $f^{-1}(x) = -\sqrt[4]{\frac{1}{x} -4}$

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