Even and Odd Functions Math Example 2

Follow the full solution, then compare it with the other examples linked below.

medium

Determine whether

g(x) = \frac{x}{x^2 + 1}

is even, odd, or neither.

1
Compute $g(-x) = \frac{-x}{(-x)^2 + 1} = \frac{-x}{x^2 + 1}$ .
2
Compare with $g(x) = \frac{x}{x^2+1}$ : we see $g(-x) = -\frac{x}{x^2+1} = -g(x)$ .
3
Since $g(-x) = -g(x)$ for all $x$ , the function is odd.

Answer

\text{Odd}

A function is odd if

f(-x) = -f(x)

for all

x

, meaning its graph has rotational symmetry about the origin (

180°

rotation). Here, the numerator is odd (

-x

) and the denominator is even (

x^2 + 1

), making the ratio odd.

About Even and Odd Functions

An even function satisfies $f(-x) = f(x)$ (symmetric about $y$ -axis); an odd function satisfies $f(-x) = -f(x)$ (rotational symmetry about origin).

Determine whether [formula] is even, odd, or neither.

Is [formula] even, odd, or neither?

Prove that the product of two odd functions is an even function.