A function $f$ is even if $f(-x) = f(x)$ for all $x$ in its domain. The graph of an even function is symmetric about the y-axis.
| Function | $f(-x)$ | Even? | Graph Symmetry |
|---|---|---|---|
| $f(x) = x^2$ | $(-x)^2 = x^2$ | β Yes | y-axis |
| $f(x) = \cos x$ | $\cos(-x) = \cos x$ | β Yes | y-axis |
| $f(x) = |x|$ | $|-x| = |x|$ | β Yes | y-axis |
| $f(x) = x^2 + 1$ | $(-x)^2 + 1 = x^2 + 1$ | β Yes | y-axis |
Determine if $f(x) = 3x^4 - 2x^2 + 5$ is even.
A function $f$ is odd if $f(-x) = -f(x)$ for all $x$ in its domain. The graph of an odd function is symmetric about the origin (rotational symmetry of $180^\circ$).
| Function | $f(-x)$ | $-f(x)$ | Odd? |
|---|---|---|---|
| $f(x) = x^3$ | $(-x)^3 = -x^3$ | $-x^3$ | β Yes |
| $f(x) = \sin x$ | $\sin(-x) = -\sin x$ | $-\sin x$ | β Yes |
| $f(x) = \frac{1}{x}$ | $\frac{1}{-x} = -\frac{1}{x}$ | $-\frac{1}{x}$ | β Yes |
| $f(x) = x^5 - x$ | $(-x)^5 - (-x) = -x^5 + x$ | $-(x^5 - x) = -x^5 + x$ | β Yes |
Determine if $f(x) = 2x^3 - x$ is odd.
Many functions are neither even nor odd. This happens when $f(-x) \neq f(x)$ and $f(-x) \neq -f(x)$.
| Function | $f(-x)$ | Even? | Odd? |
|---|---|---|---|
| $f(x) = x^2 + x$ | $x^2 - x$ | No (β $x^2 + x$) | No (β $-x^2 - x$) |
| $f(x) = e^x$ | $e^{-x}$ | No | No |
| $f(x) = \ln x$ | $\ln(-x)$ undefined | No | No |
| $f(x) = \sqrt{x}$ | $\sqrt{-x}$ undefined | No | No |
Determine if $f(x) = x^2 + 2x$ is even, odd, or neither.
Interactive graph below. Click buttons to see even functions (y-axis symmetry) and odd functions (origin symmetry):
| Property | Even Functions | Odd Functions |
|---|---|---|
| Integral over symmetric interval $[-a, a]$ | $\displaystyle \int_{-a}^{a} f(x) \, dx = 2\int_{0}^{a} f(x) \, dx$ | $\displaystyle \int_{-a}^{a} f(x) \, dx = 0$ |
| Derivative | Derivative of an even function is odd | Derivative of an odd function is even |
| Fourier Series | Only cosine terms | Only sine terms |
Evaluate $\int_{-2}^{2} x^3 \, dx$ using symmetry.
Evaluate $\int_{-2}^{2} x^2 \, dx$ using symmetry.
Determine if $f(x) = x^6 - 4x^2$ is even, odd, or neither.
$f(-x) = (-x)^6 - 4(-x)^2 = x^6 - 4x^2 = f(x)$. So $f$ is even.
Determine if $f(x) = x^5 + x^3$ is even, odd, or neither.
$f(-x) = (-x)^5 + (-x)^3 = -x^5 - x^3 = -(x^5 + x^3) = -f(x)$. So $f$ is odd.
Determine if $f(x) = x^3 + 1$ is even, odd, or neither.
$f(-x) = -x^3 + 1$. This is not equal to $f(x) = x^3 + 1$ (not even), and not equal to $-f(x) = -x^3 - 1$ (not odd). So $f$ is neither.
Evaluate $\int_{-3}^{3} \sin x \, dx$ using symmetry.
$\sin x$ is odd. For odd functions, $\int_{-a}^{a} f(x) \, dx = 0$. So $\int_{-3}^{3} \sin x \, dx = 0$.
Evaluate $\int_{-2}^{2} (x^4 + 1) \, dx$ using symmetry.
$f(x) = x^4 + 1$ is even. $\int_{-2}^{2} (x^4 + 1) dx = 2\int_{0}^{2} (x^4 + 1) dx = 2\left[\frac{x^5}{5} + x\right]_{0}^{2} = 2\left(\frac{32}{5} + 2\right) = 2\left(\frac{32}{5} + \frac{10}{5}\right) = 2 \cdot \frac{42}{5} = \frac{84}{5}$.