Even function

Definition

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

5 Must Know Facts For Your Next Test

1. An example of an even function is $f(x) = x^2$.
2. The graph of an even function remains unchanged when reflected across the y-axis.
3. If the integral of an even function over a symmetric interval around zero, like $\int_{-a}^{a} f(x) \, dx$, it can be simplified to $2 \int_{0}^{a} f(x) \, dx$.
4. The sum or difference of two even functions is also an even function.
5. Even functions often appear in trigonometric identities, such as $\cos(x)$ being an even function.

Review Questions

• What property must hold true for a function to be considered even?
• How does the graph of an even function behave with respect to the y-axis?
• Give an example of how you would simplify the integral $\int_{-a}^{a} f(x) \, dx$ if $f(x)$ is an even function.

Related terms

Odd Function: A function $f(x)$ is odd if it satisfies $f(-x) = -f(x)$ for all $x$ in its domain. Odd functions are symmetric with respect to the origin.

Symmetry:

In mathematics, symmetry refers to a situation where one part of a figure or object mirrors another part. In functions, this often refers to symmetry about the y-axis or origin.

Net Change Theorem: A principle that states that the net change in a quantity over an interval can be found using definite integrals: $\int_{a}^{b} f'(x) \, dx = f(b) - f(a)$.