To find the inverse of the function [tex]\( f(x) = 5x + 10 \)[/tex], we'll follow these steps:
1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[
y = 5x + 10
\][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
This step is crucial because the goal of finding the inverse function is to solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]. We imagine [tex]\( y \)[/tex] as the input and [tex]\( x \)[/tex] as the output.
[tex]\[
x = 5y + 10
\][/tex]
3. Solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
Isolate [tex]\( y \)[/tex] on one side of the equation:
[tex]\[
x - 10 = 5y
\][/tex]
[tex]\[
y = \frac{x - 10}{5}
\][/tex]
4. Rewrite the expression:
Here we can simplify and represent the expression:
[tex]\[
y = \frac{1}{5}x - 2
\][/tex]
Thus, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[
f^{-1}(x) = \frac{1}{5}x - 2
\][/tex]
Now, let's compare this with the given options:
- A) [tex]\( f^{-1}(x) = 10x + 5 \)[/tex]
- B) [tex]\( f^{-1}(x) = \frac{x + 10}{5} \)[/tex]
- C) [tex]\( f^{-1}(x) = 5x - 10 \)[/tex]
- D) [tex]\( f^{-1}(x) = \frac{1}{5}x - 2 \)[/tex]
Option D, [tex]\( f^{-1}(x) = \frac{1}{5}x - 2 \)[/tex], correctly matches the inverse function we found.
Therefore, the correct answer is: D: [tex]\( f^{-1}(x) = \frac{1}{5}x - 2 \)[/tex]
