Which represents the inverse of the function [tex]f(x) = 4x[/tex]?

A. [tex]h(x) = x + 4[/tex]
B. [tex]h(x) = x - 4[/tex]
C. [tex]h(x) = \frac{3}{4} x[/tex]
D. [tex]h(x) = \frac{1}{4} x[/tex]



Answer :

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]

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