To determine the inverse function of [tex]\( f(x) = \frac{1}{2}x + 5 \)[/tex], we need to follow these steps:
1. Start with the given function:
[tex]\( y = \frac{1}{2}x + 5 \)[/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex], since the inverse function essentially switches the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\( x = \frac{1}{2}y + 5 \)[/tex]
3. Solve for [tex]\( y \)[/tex] to find the inverse function:
- Subtract 5 from both sides to isolate the term involving [tex]\( y \)[/tex]:
[tex]\( x - 5 = \frac{1}{2}y \)[/tex]
- Multiply both sides of the equation by 2 to solve for [tex]\( y \)[/tex]:
[tex]\( 2(x - 5) = y \)[/tex]
- Simplify the right-hand side:
[tex]\( y = 2(x - 5) \)[/tex]
- Distribute the 2:
[tex]\( y = 2x - 10 \)[/tex]
Thus, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[ f^{-1}(x) = 2x - 10 \][/tex]
The correct option is:
[tex]\[ f^{-1}(x) = 2x - 10 \][/tex]
So, the inverse function of [tex]\( f(x) = \frac{1}{2}x + 5 \)[/tex] is [tex]\( f^{-1}(x) = 2x - 10 \)[/tex].
