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Find the inverse function of the function f, of, x, equals, cube root of, 4, x, end cube rootf(x)=

3

4x
​
.
Answer
Attempt 1 out of 2
Multiple Choice Answers
f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, start fraction, x, squared, divided by, 4, end fractionf
−1
(x)=
4
x
2

​

f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, start fraction, x, cubed, divided by, 64, end fractionf
−1
(x)=
64
x
3

​

f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, start fraction, x, cubed, divided by, 4, end fractionf
−1
(x)=
4
x
3

​

f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, start fraction, x, squared, divided by, 16, end fractionf
−1
(x)=
16
x
2

​

1 answer

Let's find the inverse of the function \( f(x) = \sqrt[3]{4x} \).

To find the inverse, follow these steps:

  1. Replace \( f(x) \) with \( y \): \[ y = \sqrt[3]{4x} \]

  2. Solve for \( x \) in terms of \( y \):

    • First, remove the cube root by cubing both sides: \[ y^3 = 4x \]

    • Now, solve for \( x \): \[ x = \frac{y^3}{4} \]

  3. Replace \( y \) with \( x \) to denote the inverse function: \[ f^{-1}(x) = \frac{x^3}{4} \]

Therefore, the inverse function is given by: \[ f^{-1}(x) = \frac{x^3}{4} \]

Now, let's check which option corresponds to this inverse function:

  • The choice \( f^{-1}(x) = \frac{x^3}{4} \) is not provided as a direct option in your choices, but there is a misunderstanding with the \( 4 \).

In this context, the correct choice based on the options provided is:

  • The option that corresponds to \( f^{-1}(x) = \frac{x^3}{4} \) is: \[ f^{-1}(x) = \frac{x^3}{4} \]

So the answer is: \[ f^{-1}(x) = 4 \cdot x^3 \quad (\text{which is not correct; should be }\frac{x^3}{4}) \]

Correctly identifying each option:

  • Which does relate as none truly matches in the original selection seen, correct statement concludes that no options are direct to outcomes achieved as any alongside must have been initially stated or referenced incorrectly.

Check through provided options for clarity with respect to \(x\)'s cubic correlation in denominative specification.

The ultimate, truly aligned option directly with calculations of inverse function as executed in workflow detail reflects: \[ \text{Exact inverse sought clearly requires re-evaluation as derived above, none listed accurately match.} \]