Question

Show that I and g are inverse functions algebraically, graphically, and numerically.

f(x) =
g(x)=√/5x
(a) algebraically
f(g(x)) =
F(
g(f(x)) = 9
=
(b) graphically
y
10
8
6
4
2
0-2
10
1
2
3
5
10
0-2
10
y
8
8

Answer 1

`\boxed{\begin{aligned}f(g(x))&=f\left(\sqrt[3]{5x}\right)\\&=x\end{aligned}}`
`\boxed{\begin{aligned}g(f(x))&=g\left((x^3)/(5)\right)\\&=x\end{aligned}}`

The graph of f is the reflection of the graph of g in the line y = x.

Explanation:

To show that two functions are inverses, we need to show that:

• f(g(x)) = x, for all x in the domain of g(x), and
• g(f(x)) = x, for all x in the domain of f(x).

Given functions:

`f(x)=(x^3)/(5)`

`g(x)=\sqrt[3]{5x}`

Find f(g(x)) by substituting x = g(x) into f(x):

`\begin{aligned}f(g(x))&=f\left(\sqrt[3]{5x}\right)\\\\&=\frac{\left(\sqrt[3]{5x}\right)^3}{5}\\\\&=\frac{\left((5x)^{(1)/(3)}\right)^3}{5}\\\\&=(5x)/(5)\\\\&=x\end{aligned}`

Find g(f(x)) by substituting x = f(x) into g(x):

`\begin{aligned}g(f(x))&=g\left((x^3)/(5)\right)\\\\&=\sqrt[3]{5\left((x^3)/(5)\right)}\\\\&=\sqrt[3]{x^3}\\\\&=(x^3)^{(1)/(3)}\\\\&=x\end{aligned}`

Therefore, as f(g(x)) = x and g(f(x)) = x, we have successfully demonstrated that f(x) and g(x) are inverses.

If we graph the two functions (see attached), we can see that:

• The graph of f is the reflection of the graph of g in the line y = x.