Given [tex]\( f(x) = 4x^4 \)[/tex], find [tex]\( f^{-1}(x) \)[/tex]. Then state whether [tex]\( f^{-1}(x) \)[/tex] is a function.

A. [tex]\( y = \pm \left( \frac{x}{4} \right)^{\frac{1}{4}} \)[/tex]; [tex]\( f^{-1}(x) \)[/tex] is a function.
B. [tex]\( y = \pm \left( \frac{x}{4} \right)^{\frac{1}{4}} \)[/tex]; [tex]\( f^{-1}(x) \)[/tex] is not a function.
C. [tex]\( y = \pm \left( \frac{x}{4} \right)^4 \)[/tex]; [tex]\( f^{-1}(x) \)[/tex] is not a function.
D. [tex]\( y = \pm \left( \frac{x}{4} \right)^4 \)[/tex]; [tex]\( f^{-1}(x) \)[/tex] is a function.



Answer :

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]

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