To determine the inverse of the function [tex]\( f(x) = 4x \)[/tex], let's proceed with the following steps:
1. Represent the function with a different variable:
Instead of [tex]\( f(x) \)[/tex], let's use [tex]\( y \)[/tex] to make it easier to manipulate. So, we start with:
[tex]\[
y = 4x
\][/tex]
2. Swap the variables:
To find the inverse, we swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. This gives us:
[tex]\[
x = 4y
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
Now, we solve this equation for [tex]\( y \)[/tex] to express it as the inverse function. Divide both sides by 4:
[tex]\[
y = \frac{x}{4}
\][/tex]
Which can also be written as:
[tex]\[
y = \frac{1}{4}x
\][/tex]
4. Write the inverse function:
The resulting equation [tex]\( y = \frac{1}{4}x \)[/tex] represents the inverse function. Therefore, the inverse function [tex]\( h(x) \)[/tex] is:
[tex]\[
h(x) = \frac{1}{4}x
\][/tex]
Looking at the given options:
- [tex]\( h(x) = x + 4 \)[/tex]
- [tex]\( h(x) = x - 4 \)[/tex]
- [tex]\( h(x) = \frac{3}{4} x \)[/tex]
- [tex]\( h(x) = \frac{1}{4} x \)[/tex]
The correct one is:
[tex]\[
h(x) = \frac{1}{4} x
\][/tex]
So, the option that represents the inverse of the function [tex]\( f(x) = 4x \)[/tex] is:
[tex]\[
h(x) = \frac{1}{4} x
\][/tex]
This corresponds to the fourth option.
