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]
