Answer:
[tex]g^{-1}(x)=f(x)=e^{\frac{x}{3}}[/tex]
Step-by-step explanation:
We have been given two functions as [tex]g(x)=\text{ln}(x^3)[/tex] and [tex]f(x)=e^{\frac{x}{3}}[/tex]. We are asked to show that both functions are inverse of each other algebraically and graphically.
Let us find inverse function of [tex]g(x)=\text{ln}(x^3)[/tex] as:
[tex]y=\text{ln}(x^3)[/tex]
Interchange x and y values:
[tex]x=\text{ln}(y^3)[/tex]
Using log property [tex]\text{ln}(a^b)=b\cdot \text{ln}(a)[/tex], we will get:
[tex]x=3\cdot \text{ln}(y)[/tex]
[tex]\frac{x}{3}=\frac{3\cdot \text{ln}(y)}{3}[/tex]
[tex]\frac{x}{3}=\text{ln}(y)[/tex]
Using log definition; If [tex]\text{log}_a(b)=c[/tex], then [tex]b=a^c[/tex], we will get:
[tex]y=e^{\frac{x}{3}}[/tex]
[tex]g^{-1}(x)=e^{\frac{x}{3}}[/tex]
Therefore, we can see that function [tex]f(x)=e^{\frac{x}{3}}[/tex] is inverse of function [tex]g(x)=\text{ln}(x^3)[/tex].
We can see that both functions are symmetric about line [tex]y=x[/tex], therefore, both functions are inverse of each other.
