1. Given: [tex]f(x)=5x+10[/tex]

Which of the following is the inverse of [tex]f(x)[/tex]?

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]



Answer :

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]