To determine the inverse of the function [tex]\( f(x) = 4x \)[/tex], we need to follow a methodical approach. Here's a step-by-step solution for finding the inverse function.
### Step 1: Express the function in terms of [tex]\( y \)[/tex]
Let [tex]\( y = f(x) \)[/tex]. Thus, we have:
[tex]\[ y = 4x \][/tex]
### Step 2: Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]
To find the inverse, we interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = 4y \][/tex]
### Step 3: Solve for [tex]\( y \)[/tex]
We now solve this equation for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{x}{4} \][/tex]
### Step 4: Write the inverse function
The expression [tex]\( y = \frac{x}{4} \)[/tex] represents the inverse function. By rewriting it in function notation, we get:
[tex]\[ h(x) = \frac{x}{4} \][/tex]
### Step 5: Identify the correct option
From 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 option that represents the inverse of the function [tex]\( f(x) = 4x \)[/tex] is:
[tex]\[ h(x) = \frac{1}{4} x \][/tex]
Therefore, the inverse function is:
[tex]\[ h(x) = \frac{1}{4} x \][/tex]
