To find the inverse of the function [tex]\( f(x) = 4x \)[/tex], we need to follow a series of steps to determine which of the provided options represents this inverse.
1. Initial Function:
[tex]\[
f(x) = 4x
\][/tex]
2. Substitute [tex]\( f(x) \)[/tex]:
Let [tex]\( y = f(x) \)[/tex], so:
[tex]\[
y = 4x
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
To find the inverse, solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[
y = 4x \implies x = \frac{y}{4}
\][/tex]
4. Express the Inverse Function:
Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] (since [tex]\( y \)[/tex] was a placeholder for [tex]\( f(x) \)[/tex]):
[tex]\[
f^{-1}(x) = \frac{x}{4} = \frac{1}{4}x
\][/tex]
Now, let's compare this inverse function with the provided 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 option must match [tex]\( f^{-1}(x) = \frac{1}{4}x \)[/tex].
Hence, the function that represents the inverse of [tex]\( f(x) = 4x \)[/tex] is:
[tex]\[
h(x) = \frac{1}{4} x
\][/tex]
