Even Function

Definition

An even function is a type of mathematical function that satisfies the condition $f(x) = f(-x)$ for all values of x in its domain. This property indicates that the function is symmetrical with respect to the y-axis, meaning that if you were to fold the graph along the y-axis, both halves would match perfectly. Even functions play a significant role in understanding transformations and behaviors of functions in various contexts.

5 Must Know Facts For Your Next Test

1. The simplest examples of even functions include basic polynomials like $f(x) = x^2$ and $f(x) = x^4$, as they both satisfy the even function condition.
2. Graphs of even functions are symmetric with respect to the y-axis, which means that if (a, b) is on the graph, then (-a, b) will also be on the graph.
3. Even functions remain unchanged under vertical shifts, but horizontal shifts will not preserve their evenness.
4. The sum or difference of two even functions is also an even function, which is useful for analyzing combinations of functions.
5. When performing transformations on an even function, such as a vertical stretch or compression, the symmetry about the y-axis is maintained.

Review Questions

• How do you determine if a function is even using its algebraic definition?
  • To determine if a function is even using its algebraic definition, you must verify if it holds true that $f(x) = f(-x)$ for all values of x in the domain. This means you take any value x, substitute it into the function to get $f(x)$, then substitute -x to find $f(-x)$. If both results are identical for every x tested within the domain, then the function is classified as even.
• Discuss how transformations can affect the properties of even functions and give an example.
  • Transformations can affect the properties of even functions in different ways. For instance, vertical shifts will not alter the evenness of a function; however, horizontal shifts may change whether or not a function remains even. For example, starting with $f(x) = x^2$, which is even, if we perform a horizontal shift by replacing x with $(x - 3)$ to get $f(x - 3) = (x - 3)^2$, this new function is still even because it satisfies $f(x - 3) = f(- (x - 3))$. However, translating horizontally does not maintain symmetry relative to the y-axis.
• Evaluate how understanding even functions enhances your ability to analyze complex graphs and their transformations.
  • Understanding even functions enhances your ability to analyze complex graphs and their transformations by providing insight into the symmetrical properties of these graphs. When you recognize that a function is even, you can predict how its graph will behave under transformations like reflections across the y-axis. This knowledge allows you to simplify analysis since you only need to consider one side of the graph and reflect it to understand its entirety. Moreover, it aids in predicting outcomes when combining functions; knowing that two even functions sum to another even function can streamline calculations in various problems.

Related terms

Odd Function:

An odd function is defined by the condition $f(-x) = -f(x)$ for all x in its domain, indicating symmetry about the origin.

Symmetry:

Symmetry in mathematics refers to a situation where a figure or graph remains unchanged when subjected to certain transformations, such as reflection or rotation.

Function Transformation: Function transformation involves changing the position, shape, or size of a function's graph through operations like translation, reflection, stretching, or compression.