To find the inverse function [tex]\(f^{-1}(x)\)[/tex] for the given function [tex]\(f(x)=4x^4\)[/tex], follow these steps:
1. Express the function analytically:
Given:
[tex]\[
f(x) = 4x^4
\][/tex]
2. Let [tex]\(y\)[/tex] represent [tex]\(f(x)\)[/tex]:
Thus:
[tex]\[
y = 4x^4
\][/tex]
3. Switch [tex]\(x\)[/tex] and [tex]\(y\)[/tex] to find the inverse:
Switch [tex]\(x\)[/tex] and [tex]\(y\)[/tex] to solve for the inverse function:
[tex]\[
x = 4y^4
\][/tex]
4. Solve for [tex]\(y\)[/tex]:
Divide both sides by 4:
[tex]\[
\frac{x}{4} = y^4
\][/tex]
Take the fourth root of both sides:
[tex]\[
y = \pm\left(\frac{x}{4}\right)^{\frac{1}{4}}
\][/tex]
Hence, we have:
[tex]\[
f^{-1}(x) = \pm\left(\frac{x}{4}\right)^{\frac{1}{4}}
\][/tex]
5. Determine if [tex]\(f^{-1}(x)\)[/tex] is a function:
The expression [tex]\( f^{-1}(x) = \pm\left(\frac{x}{4}\right)^{\frac{1}{4}} \)[/tex] indicates that for each [tex]\(x\)[/tex], there are two possible values for [tex]\(y\)[/tex]: one positive and one negative. Because an inverse function must assign exactly one output to each input, this does not satisfy the requirement of being a function (fails the vertical line test).
Hence,
- The correct form is [tex]\( y = \pm\left(\frac{x}{4}\right)^{\frac{1}{4}} \)[/tex].
- And this is not a function.
Therefore, the final answer is:
[tex]\[
\boxed{y= \pm\left(\frac{x}{4}\right)^{\frac{1}{4}} ; f^{-1}(x)\text{ is not a function.}}
\][/tex]
