In Target B, we analyzed polynomial functions. This included finding the degree of the function, determining the end behavior of the graph, finding the maximum or minimum values of the function (and where they occur), finding the intervals where the function increases and decreases, finding the y-intercept, and finding the x-intercepts, including their multiplicity.
But there is one more thing that can be analyzed about functions, and that is to determine if the function is an even function, an odd function or neither even nor odd. We will only address this concept in the context of polynomial functions, but this concept is applicable to any function and will be addressed in future math courses.
Do not confuse even and odd functions with the degree of the function. Although it is true that an even polynomial function must have an even degree, and an odd polynomial function must have an odd degree, not all even degree functions will be even and not all odd degree functions will be odd.
Before proceeding with the formal definition of an even or odd function, investigate the graphs below and answer the questions.
But there is one more thing that can be analyzed about functions, and that is to determine if the function is an even function, an odd function or neither even nor odd. We will only address this concept in the context of polynomial functions, but this concept is applicable to any function and will be addressed in future math courses.
Do not confuse even and odd functions with the degree of the function. Although it is true that an even polynomial function must have an even degree, and an odd polynomial function must have an odd degree, not all even degree functions will be even and not all odd degree functions will be odd.
Before proceeding with the formal definition of an even or odd function, investigate the graphs below and answer the questions.
Quick Check 1
1. What type of symmetry exists?
2. When the \(x\) values (inputs) are opposite in sign, what do you notice about the \(y\) values (outputs)?
Quick Check Solutions
Quick Check 2
1. What type of symmetry exists?
2. When the \(x\) values (inputs) are opposite in sign, what do you notice about the \(y\) values (outputs)?
Quick Check Solutions
The first graph (top) represents an EVEN function and the second graph (bottom) represents an ODD function.
Even Functions
Odd Functions
If a function does not satisfy the criteria to be even or odd, we say the function is neither even nor odd.
Example 1: Given the graphs, determine whether the function is even, odd or neither.
Even Functions
- A function is even if the graph has reflective symmetry over the y-axis.
- A function is even if \(f\left(-x\right)=f\left(x\right)\). This means if you evaluate the function at opposite inputs, you will get the same outputs. For example, if you evaluated a function at \(x=2\) and \(x=-2\), you would get the same \(y\) value.
Odd Functions
- A function is odd if the graph has rotational symmetry about the origin, \(\left(0,0\right)\).
- A function is odd if \(f\left(-x\right)=-f\left(x\right)\). This means if you evaluate the function at opposite inputs, you will get opposite outputs. For example, if you evaluated a function at \(x= 2\) and \(x=-2\), you would get opposite \(y\) values.
If a function does not satisfy the criteria to be even or odd, we say the function is neither even nor odd.
Example 1: Given the graphs, determine whether the function is even, odd or neither.
Graph \(a\) is even because it has reflective symmetry with respect to the y-axis. Graph \(b\) is neither even nor odd because it does not have reflective symmetry with the y-axis or rotational symmetry about the origin. Graph \(c\) is odd because it has rotational symmetry about the origin.
To determine algebraically if a graph is even, odd, or neither, follow the steps below:
1. Find \(f\left(-x\right)\) and compare it to the original function \(f\left(x\right)\).
If \(f\left(-x\right)=f\left(x\right)\) then the function is even (and there is no more work to do!)
2. If the graph is not even, find \(-f\left(x\right)\) and compare it to \(f\left(-x\right)\).
If \(f\left(-x\right)=-f\left(x\right)\) then the function is odd.
3. If \(f\left(-x\right)\ne f\left(x\right)\) (that is, the function is not even) AND \(f\left(-x\right)\ne-f\left(x\right)\) (that is, the function is not odd), we
determine that the function is neither even nor odd.
Example 2: Determine algebraically if the function \(f\left(x\right)=4x^3-2x+1\) is even, odd, or neither.
1. Find \(f\left(-x\right)\)
\(\begin{align}&\left(-x\right)=4\left(-x\right)^3-2\left(-x\right)+1\ & \ \ & \text{Substitute -x in for x.}\\
&\left(-x\right)=-4x^3+2x+1\ & \ \ & \text{Remember that a negative value raised to an odd power is negative}\\
&\ & \ \ &\text{and a negative multiplied with a negative is positive.}\end{align}\)
Since \(-4x^3+2x+1\ne4x^3-2x+1\) (that is, \(f\left(-x\right)\ne f\left(x\right)\)) the function is not even.
2. Find \(-f\left(x\right)\)
\(\begin{align}&-f\left(x\right)=-\left(4x^3-2x+1\right)\ & \ \ & \text{Evaluate the opposite of the function.}\\
&-f\left(x\right)=-4x^3+2x-1\ & \ \ & \text{Distribute the negative.}\end{align}\)
Since \(-4x^3+2x+1\ne-4x^3+2x-1\) (that is, \(f\left(-x\right)\ne-f\left(x\right)\)) the function is not odd.
Therefore \(f\left(x\right)=4x^3-2x+1\) is neither even nor odd.
Example 3: Determine algebraically if the function \(f\left(x\right)=2x^4+x^2-1\) is even, odd, or neither.
Example 4: Determine algebraically if the function \(f\left(x\right)=x^5+3x\) is even, odd, or neither.