To determine the inverse of the function [tex]\( f(x) = 4x \)[/tex], we need to follow these steps:
1. Rewrite the function using [tex]\( y \)[/tex] instead of [tex]\( f(x) \)[/tex]:
[tex]\[
y = 4x
\][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to reflect the inverse relationship:
[tex]\[
x = 4y
\][/tex]
3. Solve this new equation for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{x}{4}
\][/tex]
4. Express the inverse function: In terms of function notation, the inverse function can be written as:
[tex]\[
h(x) = \frac{x}{4}
\][/tex]
Upon simplifying, this is equivalent to:
[tex]\[
h(x) = \frac{1}{4}x
\][/tex]
Therefore, the inverse of the function [tex]\( f(x) = 4x \)[/tex] is represented by:
[tex]\[
h(x) = \frac{1}{4}x
\][/tex]
From the given options, the correct representation of the inverse function is:
[tex]\[
h(x) = \frac{1}{4}x
\][/tex]
So the correct answer is:
[tex]\[
h(x) = \frac{1}{4}x
\][/tex]
