Chapter 0, Problem 2RWDT

To determine

To find: if the composite function $f\circ g$ is even or odd.

Answer to Problem 2RWDT

Even Function

Explanation of Solution

Given information:

It is given that f and g are functions with domain all real and f is an odd function and g is an even function.

Definition Used:

A function $y=f\left(x\right)$ is an

Even function of x if $f\left(-x\right)=f\left(x\right)$

Odd function of x if $f\left(-x\right)=-f\left(x\right)$

For every x in the function’s domain.

It is given that f is an odd function and g is an even function, so by the above definition,

  $f\left(-x\right)=-f\left(x\right)$

  $g\left(-x\right)=g\left(x\right)$

Now, consider the composite function $f\circ g$ . Since the domain of both f and g is all real, so domain of the composite function is also all reals.

Now, for any $x\in ℝ$ , the composite function is defined as:

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

Checking if function $f\circ g$ is even or odd.

  $\begin{array}{c}f\circ g\left(-x\right)=f\left(g\left(-x\right)\right)\\ =f\left(g\left(x\right)\right)\left[\because \text{ g is even}\right]\\ =f\circ g\left(x\right)\end{array}$

Hence, the function $f\circ g$ is even.