Q. 76 - Precalculus Enhanced with Graphing Utilities

Question

If $f$ and $g$ are odd functions, show that the composite function $f\circ g$ is also odd. 

Step-by-Step Solution

Answer

The, $f\circ g$ is an odd function.

Step 1. Given Information

Given that $f$ and $g$ are odd functions.

Step 2. Explanation

A function $f$ is said to be odd function if $f(-x)=-f\left(x\right), \forall x$.

Since, $f$ and $g$ are odd function, So

$$\begin{gathered} f(-x)=-f\left(x\right) \\ g(-x)=-g\left(x\right) \end{gathered}$$

Now,

$f\circ g\left(x\right)=f\left(g\right(x\left)\right)$

then,

$$\begin{gathered} f\circ g(-x)=f\left(g\right(-x\left)\right) \\ f\circ g(-x)=f(-g(x\left)\right) \{g is odd function.\} \\ f\circ g(-x)=-f\left(g\right(x\left)\right) \{f is odd function.\} \\ f\circ g(-x)=-f\circ g\left(x\right) \forall x \end{gathered}$$

Hence, $f\circ g$ is an odd function.