To determine the inverse of the function [tex]\( f(x) = 4x \)[/tex], let's go through the steps for finding an inverse function. Here are the steps:
1. Substitute [tex]\( y \)[/tex] for [tex]\( f(x) \)[/tex]:
[tex]\[
y = 4x
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
To find the inverse, we need to express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]. Starting with the equation [tex]\( y = 4x \)[/tex], we solve for [tex]\( x \)[/tex] by dividing both sides by 4:
[tex]\[
x = \frac{y}{4}
\][/tex]
3. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
Since [tex]\( y \)[/tex] was originally [tex]\( f(x) \)[/tex], when we find the inverse, we swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
y = \frac{x}{4}
\][/tex]
4. Write the inverse function:
The resulting expression [tex]\( y = \frac{x}{4} \)[/tex] represents the inverse function formula. Thus, the inverse function [tex]\( h(x) \)[/tex] is:
[tex]\[
h(x) = \frac{1}{4}x
\][/tex]
Therefore, the inverse function of [tex]\( f(x) = 4x \)[/tex] is given by:
[tex]\[
h(x) = \frac{1}{4}x
\][/tex]
Comparing this with the given options, the correct answer is:
[tex]\[
h(x) = \frac{1}{4}x
\][/tex]
