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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]


Sagot :

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.