To find the inverse function of the given function [tex]\( f(x) = \frac{x}{2} + 8 \)[/tex], follow these steps:
1. Rewrite the function using a different variable: Let [tex]\( y = f(x) \)[/tex]. This gives us the equation:
[tex]\[
y = \frac{x}{2} + 8
\][/tex]
2. Express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]: To find the inverse function, we need to solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]. Start by isolating [tex]\( x \)[/tex] on one side of the equation:
[tex]\[
y = \frac{x}{2} + 8
\][/tex]
3. Subtract 8 from both sides:
[tex]\[
y - 8 = \frac{x}{2}
\][/tex]
4. Multiply both sides by 2 to solve for [tex]\( x \)[/tex]:
[tex]\[
2(y - 8) = x
\][/tex]
5. Simplify the expression:
[tex]\[
x = 2y - 16
\][/tex]
Since we want the inverse function [tex]\( f^{-1}(x) \)[/tex], we express this result with [tex]\( x \)[/tex] as the input variable and rename [tex]\( y \)[/tex] back to [tex]\( x \)[/tex]:
[tex]\[
f^{-1}(x) = 2x - 16
\][/tex]
Therefore, the inverse function is:
[tex]\[
f^{-1}(x) = 2x - 16
\][/tex]
