Even function

Definition

An even function is a function $f(x)$ such that $f(x) = f(-x)$ for all $x$ in its domain. Graphically, even functions are symmetrical with respect to the y-axis.

5 Must Know Facts For Your Next Test

1. If $f(x)$ is an even function and integrable on $[-a, a]$, then $\int_{-a}^{a} f(x) \, dx = 2 \int_{0}^{a} f(x) \, dx$.
2. The power functions $x^{2n}$ (where $n$ is an integer) are examples of even functions.
3. Even functions can simplify integrals over symmetric intervals by doubling the integral from 0 to $a$.
4. Cosine function, $\cos(x)$, is an example of an even trigonometric function.
5. If a polynomial contains only even powers of x (e.g., $x^2$, $x^4$), it is an even function.

Review Questions

• What property must an even function satisfy for all x in its domain?
• How does the symmetry of an even function affect its integral over symmetric limits?
• Give an example of a polynomial that represents an even function.

Related terms

Odd Function: A function $f(x)$ such that $f(-x) = -f(x)$ for all $x$ in its domain.

$\cos(x)$: The cosine function, which is an example of an even trigonometric function.

$\int_{-a}^{a} f(x) \, dx$: Integral from -a to a; if the integrand is an even function, it equals twice the integral from 0 to a.