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What is the inverse of the function [tex]$f(x)=\frac{1}{4}x-12$[/tex]?

A. [tex]h(x)=48x-4[/tex]

B. [tex]h(x)=48x+4[/tex]

C. [tex]h(x)=4x-48[/tex]

D. [tex]h(x)=4x+48[/tex]


Sagot :

To find the inverse of the function [tex]\( f(x) = \frac{1}{4}x - 12 \)[/tex], follow these steps:

1. Rewrite the function as [tex]\( y = \frac{1}{4}x - 12 \)[/tex]:
[tex]\[ y = \frac{1}{4}x - 12 \][/tex]

2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse function:
[tex]\[ x = \frac{1}{4}y - 12 \][/tex]

3. Solve the equation for [tex]\( y \)[/tex]:
- First, add 12 to both sides:
[tex]\[ x + 12 = \frac{1}{4}y \][/tex]

- Next, multiply both sides by 4 to isolate [tex]\( y \)[/tex]:
[tex]\[ 4(x + 12) = y \][/tex]
[tex]\[ y = 4x + 48 \][/tex]

4. Therefore, the inverse function [tex]\( h(x) \)[/tex] is:
[tex]\[ h(x) = 4x + 48 \][/tex]

Given the options:
- [tex]\( h(x) = 48x - 4 \)[/tex]
- [tex]\( h(x) = 48x + 4 \)[/tex]
- [tex]\( h(x) = 4x - 48 \)[/tex]
- [tex]\( h(x) = 4x + 48 \)[/tex]

The correct inverse function is:
[tex]\[ h(x) = 4x + 48 \][/tex]

So the correct option is:
[tex]\[ h(x)=4x+48 \][/tex]