Let's match the given functions with their inverse functions step-by-step.
1. For the function [tex]\( f(x)=\frac{2x}{3}-17 \)[/tex]:
To find the inverse, we need an expression such that when we apply [tex]\( f \)[/tex] to it, we get [tex]\( x \)[/tex] back.
Given answer:
[tex]\[ f^{-1}(x) = \frac{3(x+17)}{2} \][/tex]
So, the inverse function for [tex]\( f(x)=\frac{2x}{3}-17 \)[/tex] is:
[tex]\[ f^{-1}(x) = \frac{3(x+17)}{2} \][/tex]
2. For the function [tex]\( f(x)=x-10 \)[/tex]:
Given answer:
[tex]\[ f^{-1}(x) = x + 10 \][/tex]
So, the inverse function for [tex]\( f(x)=x-10 \)[/tex] is:
[tex]\[ f^{-1}(x) = x + 10 \][/tex]
3. For the function [tex]\( f(x)=\sqrt[3]{2x} \)[/tex]:
Given answer:
[tex]\[ f^{-1}(x) = \frac{x^3}{2} \][/tex]
So, the inverse function for [tex]\( f(x)=\sqrt[3]{2x} \)[/tex] is:
[tex]\[ f^{-1}(x) = \frac{x^3}{2} \][/tex]
4. For the function [tex]\( f(x)=\frac{x}{5} \)[/tex]:
Given answer:
[tex]\[ f^{-1}(x) = 5x \][/tex]
So, the inverse function for [tex]\( f(x)=\frac{x}{5} \)[/tex] is:
[tex]\[ f^{-1}(x) = 5x \][/tex]
Now let's put it all together:
[tex]\[
\begin{array}{c|c}
\text{Function} & \text{Inverse Function} \\
\hline
f(x)=\frac{2x}{3}-17 & f^{-1}(x)=\frac{3(x + 17)}{2} \\
f(x)=x-10 & f^{-1}(x)=x+10 \\
f(x)=\sqrt[3]{2x} & f^{-1}(x)=\frac{x^3}{2} \\
f(x)=\frac{x}{5} & f^{-1}(x)=5x \\
\end{array}
\][/tex]
