What is the inverse of the function [tex]f(x) = 2x - 10[/tex]?

A. [tex]h(x) = 2x - 5[/tex]
B. [tex]h(x) = 2x + 5[/tex]
C. [tex]h(x) = \frac{1}{2}x - 5[/tex]
D. [tex]h(x) = \frac{1}{2}x + 5[/tex]



Answer :

To find the inverse of the function [tex]\(f(x) = 2x - 10\)[/tex], follow these steps:

1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = 2x - 10 \][/tex]

2. Interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex]: This step is because the inverse function [tex]\( f^{-1}(x) \)[/tex] is obtained by swapping the dependent and independent variables.
[tex]\[ x = 2y - 10 \][/tex]

3. Solve for [tex]\( y \)[/tex]:
- Add 10 to both sides to isolate the [tex]\( y \)[/tex]-term:
[tex]\[ x + 10 = 2y \][/tex]
- Divide both sides by 2 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{x + 10}{2} \][/tex]

4. Simplify the expression:
- Split the fraction:
[tex]\[ y = \frac{x}{2} + \frac{10}{2} \][/tex]
- Simplify the right-hand side:
[tex]\[ y = \frac{1}{2}x + 5 \][/tex]

Thus, the inverse function of [tex]\( f(x) = 2x - 10 \)[/tex] is:
[tex]\[ h(x) = \frac{1}{2}x + 5 \][/tex]

Therefore, the correct answer is:
[tex]\[ h(x) = \frac{1}{2}x + 5 \][/tex]

Hence, among the given options:
- [tex]\( h(x) = 2x - 5 \)[/tex]
- [tex]\( h(x) = 2x + 5 \)[/tex]
- [tex]\( h(x) = \frac{1}{2}x - 5 \)[/tex]
- [tex]\( h(x) = \frac{1}{2}x + 5 \)[/tex]

The correct choice is:
[tex]\[ h(x) = \frac{1}{2}x + 5 \][/tex]