To find the inverse of the function [tex]\( f(x) = 4x \)[/tex], we need to follow certain steps. Here is a detailed, step-by-step solution:
1. Understand the Definition of an Inverse Function:
The inverse of a function [tex]\( f(x) \)[/tex] is a function that reverses the effect of [tex]\( f \)[/tex]. If we denote [tex]\( f \)[/tex] as [tex]\( f(x) \)[/tex] and its inverse as [tex]\( h(x) \)[/tex], then [tex]\( h \)[/tex] should satisfy the condition [tex]\( h(f(x)) = x \)[/tex] for all [tex]\( x \)[/tex] in the domain of [tex]\( f \)[/tex].
2. Express the Function:
The given function is [tex]\( f(x) = 4x \)[/tex].
3. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
Let [tex]\( y = f(x) \)[/tex]. So, we write:
[tex]\[
y = 4x
\][/tex]
4. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
To find the inverse, we need to express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[
y = 4x \\
\Rightarrow x = \frac{y}{4}
\][/tex]
5. Rewrite the Expression as the Inverse Function:
We now have [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]. To find the inverse function [tex]\( h(x) \)[/tex], we replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] in the expression:
[tex]\[
x = \frac{y}{4} \\
\Rightarrow h(x) = \frac{x}{4}
\][/tex]
6. Identify the Correct Option:
After finding [tex]\( h(x) = \frac{x}{4} \)[/tex], we look at the given options to see which one matches our derived inverse function:
- [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 representation of the inverse function [tex]\( h(x) \)[/tex] is:
[tex]\[
h(x) = \frac{1}{4}x
\][/tex]
Thus, the option that represents the inverse of the function [tex]\( f(x) = 4x \)[/tex] is:
[tex]\[ \boxed{h(x) = \frac{1}{4}x} \][/tex]
Therefore, the correct answer is [tex]\( h(x) = \frac{1}{4}x \)[/tex].
