Chapter 4.6, Problem 64E

To determine

To find : whether the function is even, odd, or neither using the graph of the function

Answer to Problem 64E

The function $g(x)=x^2\cot x$ is $\underset{\_}{\boxed{ODD}}function$

Explanation of Solution

Given information: $g(x)=x^2\cot x$

Concept Involved:

A function is called an even function if for every input x , $g\left(-x\right)=g\left(x\right)$ . If graph of a function is symmetric about the y-axis, then the function is an even function.

A function is called an odd function if for every input x , $g\left(-x\right)=-g\left(x\right)$ . If graph of a function is symmetric about the origin, then the function is an odd function.

Graph:

  

Interpretation:

The graph of a function $g(x)=x^2\cot x$ is symmetric about the y-axis, so the function is an even function.

Calculation:

Substitute − x for x in the function to find $g\left(-x\right)$

  $g(-x)={\left(-x\right)}^2\cot \left(-x\right)$

Rewrite the function using the trigonometric identify $\cot \left(-x\right)=-\cot x$

  $g\left(-x\right)=x^2\left(-\cot x\right)$

Multiplying in right side of the equation

  $g\left(-x\right)=-x^2\cot x$▶1^st equation

Finding $-g\left(x\right)$

  $-g(x)=-x^2\cot x$▶2^ndequation

From 1^st equation and 2^nd equation, we see that $g\left(-x\right)=-g\left(x\right)$ , so we can conclude that $g(x)=x^2\cot x$ is an odd function.