Chapter 12, Problem 1CR

To determine

Whether the statement “The functions $f\left(x\right)=x^2-1\text{ and g}\left(x\right)=x^5$ are orthogonal on the interval” is true or false.

Answer to Problem 1CR

The statement is true.

Explanation of Solution

Given:

The functions are $f\left(x\right)=x^2-1\text{ and g}\left(x\right)=x^5$ .

Concept:

The functions $f\left(x\right)\text{ and g}\left(x\right)$ are said to be orthogonal on an interval $\left[a,b\right]$ if

${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)g\left(x\right)}dx=0$ .

Calculation:

The provided functions are $f\left(x\right)=x^2-1\text{ and g}\left(x\right)=x^5$,

$\begin{array}{c}{\displaystyle \underset{-\pi }{\overset{\pi }{\int }}f\left(x\right)g\left(x\right)}dx={\displaystyle \underset{-\pi }{\overset{\pi }{\int }}\left(x^2-1\right)\cdot x^5}dx\\ ={\displaystyle \underset{-\pi }{\overset{\pi }{\int }}{x}^{7}dx}-{\displaystyle \underset{-\pi }{\overset{\pi }{\int }}{x}^5}dx\end{array}$

Since, both integrands are odd so integral of odd function is zero.

${\displaystyle \underset{-\pi }{\overset{\pi }{\int }}f\left(x\right)g\left(x\right)}dx=0$

Thus, the functions $f\left(x\right)\text{ and g}\left(x\right)$ are said to be orthogonal on an interval $\left[-\pi ,\pi \right]$.

Therefore, the statement is true.