To determine the inverse of the function [tex]\( f(x) = \frac{x}{4} - 2 \)[/tex], let's follow these steps:
1. Rewrite the function:
[tex]\[
y = \frac{x}{4} - 2
\][/tex]
Here, we're expressing [tex]\( f(x) \)[/tex] as [tex]\( y \)[/tex].
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x = \frac{y}{4} - 2
\][/tex]
This step reflects the definition of the inverse function, where the input [tex]\( x \)[/tex] of the original function becomes the output of the inverse function [tex]\( f^{-1}(x) \)[/tex], and vice versa.
3. Solve for [tex]\( y \)[/tex]:
- First, isolate [tex]\( \frac{y}{4} \)[/tex]:
[tex]\[
x + 2 = \frac{y}{4}
\][/tex]
- Then, multiply both sides by 4 to solve for [tex]\( y \)[/tex]:
[tex]\[
y = 4(x + 2)
\][/tex]
4. Express the inverse function:
[tex]\[
f^{-1}(x) = 4(x + 2)
\][/tex]
So the inverse of the function [tex]\( f(x) = \frac{x}{4} - 2 \)[/tex] is:
[tex]\[
f^{-1}(x) = 4(x + 2)
\][/tex]
Thus, the correct answer is [tex]\( \boxed{4(x + 2)} \)[/tex].
To verify, let's check which option matches our result:
- A. [tex]\( f^{-1}(x) = 4(x-2) \)[/tex] (No match)
- B. [tex]\( f^{-1}(x) = 4(x+2) \)[/tex] (Match)
- C. [tex]\( f^{-1}(x) = 2(x-4) \)[/tex] (No match)
- D. [tex]\( f^{-1}(x) = 2(x+4) \)[/tex] (No match)
Based on our calculations, option B [tex]\( (4(x + 2)) \)[/tex] is the correct inverse function of [tex]\( f(x) = \frac{x}{4} - 2 \)[/tex].
