Question

Inverse Function In Exercise,analytically show that the functions are inverse functions.Then use the graphing utility to show this graphically.
f(x) = e^x/3
g(x) = Inx^3

Answer 1

Answer:

$g^{-1}(x)=f(x)=e^{\frac{x}{3}}$

Step-by-step explanation:

We have been given two functions as $g(x)=\text{ln}(x^3)$ and $f(x)=e^{\frac{x}{3}}$. We are asked to show that both functions are inverse of each other algebraically and graphically.

Let us find inverse function of $g(x)=\text{ln}(x^3)$ as:

$y=\text{ln}(x^3)$

Interchange x and y values:

$x=\text{ln}(y^3)$

Using log property $\text{ln}(a^b)=b\cdot \text{ln}(a)$, we will get:

$x=3\cdot \text{ln}(y)$

$\frac{x}{3}=\frac{3\cdot \text{ln}(y)}{3}$

$\frac{x}{3}=\text{ln}(y)$

Using log definition; If $\text{log}_a(b)=c$, then $b=a^c$, we will get:

$y=e^{\frac{x}{3}}$

$g^{-1}(x)=e^{\frac{x}{3}}$

Therefore, we can see that function $f(x)=e^{\frac{x}{3}}$ is inverse of function $g(x)=\text{ln}(x^3)$.

We can see that both functions are symmetric about line $y=x$, therefore, both functions are inverse of each other.