Question

Hello, I was wondering if you could help me see if f(x) = 2/5 x + 1/3 and g(x) = 5/2 x - 5/6 are inverses of each other.

Answer 1

Given,

The expression of the function f(x) is 2/5 x + 1/3.

The expression of the function g(x) is 5/2 x - 5/6.

Required

To identify whether the function are inverse of each other.

The function is inverse of each other when f(g(x)) = x and g(f(x)) = x.

So,

$\begin{gathered} f(g(x))=f(\frac{5}{2}x-\frac{5}{6}) \\ =\frac{2}{5}(\frac{5}{2}x-\frac{5}{6})+\frac{1}{3} \\ =\frac{2}{5}\times\frac{5}{2}x-\frac{2}{5}\times\frac{5}{6}+\frac{1}{3} \\ =x-\frac{1}{3}+\frac{1}{3} \\ =x \end{gathered}$

Checking for g(f(X)).

$\begin{gathered} g(f(x))=g(\frac{2}{5}x+\frac{1}{3}) \\ =\frac{5}{2}(\frac{2}{5}x+\frac{1}{3})-\frac{5}{6} \\ =\frac{5}{2}\times\frac{2}{5}x+\frac{5}{2}\times\frac{1}{3}-\frac{5}{6} \\ =x+\frac{5}{6}-\frac{5}{6} \\ =x \end{gathered}$

Hence, both the function are inverse of each other.