Question Type 3: Graphing odd, even or neither functions Exercises

Determine whether the function $f(x)=x^3+x$ is even, odd, or neither. Sketch its graph indicating the symmetry.

Determine whether the function $f(x)=x^4-2x^2+1$ is even, odd, or neither. Sketch its graph indicating the symmetry.

Determine whether the function $f(x)=2x^3-3x^2+x-5$ is even, odd, or neither. Sketch its graph indicating the symmetry, if any.

Determine whether the function $f(x)=\dfrac{x^3+x}{x^2-1}$ is even, odd, or neither. Justify your answer.

Determine whether the function $f(x)=e^x+e^{-x}$ is even, odd, or neither. Justify your answer.

Determine whether the function $f(x)=x^2\,\sin x$ is even, odd, or neither. Justify your answer.

Determine whether the function $f(x)=\dfrac{x^3+x}{x^2+1}$ is even, odd, or neither. Justify your answer.

Let $f$ and $g$ be odd functions. Determine whether the functions

(i) $f(x)+g(x)$, (ii) $f(x)\cdot g(x)$, (iii) $f\bigl(g(x)\bigr)$

are even or odd. Justify each answer.

Determine whether the function $f(x)=\dfrac{x}{1+x^2}$ is even, odd, or neither. Justify your answer.

Determine whether the function $f(x)=|x|^3$ is even, odd, or neither. Justify your answer.