Even Function

Definition

An even function is a type of mathematical function that exhibits symmetry about the y-axis. This means that for any input value $$x$$, the output is the same when the input is the negative of that value; in other words, $$f(-x) = f(x)$$ for all $$x$$ in the domain of the function. The property of being even leads to unique characteristics when graphing, as it results in a mirror image on either side of the y-axis.

5 Must Know Facts For Your Next Test

1. A common example of an even function is $$f(x) = x^2$$, which has a parabolic shape symmetric about the y-axis.
2. Even functions only contain even-powered terms when expressed in polynomial form, while odd functions contain odd-powered terms.
3. To determine if a function is even, you can test whether substituting $$-x$$ into the function yields the same result as substituting $$x$$.
4. Graphically, if you fold the graph along the y-axis, both halves will align perfectly for an even function.
5. Even functions can be continuous or discontinuous; however, if they are continuous, they will always exhibit the same symmetry around the y-axis.

Review Questions

• What are the characteristics that define an even function, and how can you identify it from its graph?
  • An even function is defined by the property that $$f(-x) = f(x)$$ for all values in its domain. To identify it from its graph, look for symmetry about the y-axis. If you can fold the graph along this axis and both sides match perfectly, then it's an even function. Additionally, common features include having only even-powered terms in polynomial expressions.
• How do even functions differ from odd functions regarding their algebraic properties and graphical representation?
  • Even functions are defined by the equation $$f(-x) = f(x)$$ and exhibit symmetry about the y-axis, while odd functions follow $$f(-x) = -f(x)$$ and exhibit symmetry about the origin. Algebraically, even functions only include even powers of x in their terms. Graphically, an even function's left side mirrors its right side around the y-axis, whereas an odd function's graph reflects across both axes.
• Evaluate how understanding even functions impacts the analysis of complex functions and their graphs.
  • Understanding even functions allows for deeper insight into complex functions by identifying symmetries that can simplify computations and predictions about behavior. For instance, knowing that a function is even means certain calculations can be reduced since only half of the graph needs to be evaluated to infer properties about the entire function. This concept becomes particularly useful in calculus and mathematical modeling, where symmetry can lead to simplified integrals or differential equations.

Related terms

Odd Function: An odd function is a function where the output is the negative of the input when reflected across the origin, meaning $$f(-x) = -f(x)$$.

Symmetry: Symmetry in functions refers to a property where a function exhibits balanced features, which can be about the y-axis (even), x-axis, or origin (odd).

Graph of a Function: The graph of a function visually represents its input-output relationship on a coordinate plane, allowing for analysis of properties like symmetry.