To solve this, we need to pair each given function [tex]\( f(x) \)[/tex] with its corresponding inverse function [tex]\( f^{-1}(x) \)[/tex]. Let's go through each function carefully:
1. Function: [tex]\( f(x)=\frac{2x}{3} - 17 \)[/tex]
- To find the inverse, let's solve for [tex]\( x \)[/tex]:
[tex]\[
y = \frac{2x}{3} - 17
\][/tex]
[tex]\[
y + 17 = \frac{2x}{3}
\][/tex]
[tex]\[
3(y + 17) = 2x
\][/tex]
[tex]\[
x = \frac{3(y + 17)}{2}
\][/tex]
Thus, the inverse function is [tex]\( f^{-1}(x) = \frac{3(x + 17)}{2} \)[/tex].
2. Function: [tex]\( f(x)=x - 10 \)[/tex]
- To find the inverse, let's solve for [tex]\( x \)[/tex]:
[tex]\[
y = x - 10
\][/tex]
[tex]\[
y + 10 = x
\][/tex]
Thus, the inverse function is [tex]\( f^{-1}(x) = x + 10 \)[/tex].
3. Function: [tex]\( f(x)=\sqrt[3]{2x} \)[/tex]
- To find the inverse, let's solve for [tex]\( x \)[/tex]:
[tex]\[
y = \sqrt[3]{2x}
\][/tex]
[tex]\[
y^3 = 2x
\][/tex]
[tex]\[
x = \frac{y^3}{2}
\][/tex]
Thus, the inverse function is [tex]\( f^{-1}(x) = \frac{x^3}{2} \)[/tex].
4. Function: [tex]\( f(x)=\frac{x}{5} \)[/tex]
- To find the inverse, let's solve for [tex]\( x \)[/tex]:
[tex]\[
y = \frac{x}{5}
\][/tex]
[tex]\[
y \cdot 5 = x
\][/tex]
Thus, the inverse function is [tex]\( f^{-1}(x) = 5x \)[/tex].
Now let's match each function with its inverse function using the detailed steps:
- [tex]\( f(x) = \frac{2x}{3} - 17 \)[/tex] corresponds to [tex]\( f^{-1}(x) = \frac{3(x + 17)}{2} \)[/tex]
- [tex]\( f(x) = x - 10 \)[/tex] corresponds to [tex]\( f^{-1}(x) = x + 10 \)[/tex]
- [tex]\( f(x) = \sqrt[3]{2x} \)[/tex] corresponds to [tex]\( f^{-1}(x) = \frac{x^3}{2} \)[/tex]
- [tex]\( f(x) = \frac{x}{5} \)[/tex] corresponds to [tex]\( f^{-1}(x) = 5x \)[/tex]
Therefore:
Function [tex]\( \quad \quad \longrightarrow \)[/tex] Inverse Function
[tex]\( f(x)=\frac{2x}{3} - 17 \)[/tex] [tex]\( \longrightarrow \)[/tex] [tex]\( f^{-1}(x)=\frac{3(x+17)}{2} \)[/tex]
[tex]\( f(x)=x - 10 \)[/tex] [tex]\( \longrightarrow \)[/tex] [tex]\( f^{-1}(x)=x + 10 \)[/tex]
[tex]\( f(x)=\sqrt[3]{2x} \)[/tex] [tex]\( \longrightarrow \)[/tex] [tex]\( f^{-1}(x)=\frac{x^3}{2} \)[/tex]
[tex]\( f(x)=\frac{x}{5} \)[/tex] [tex]\( \longrightarrow \)[/tex] [tex]\( f^{-1}(x)=5x \)[/tex]
