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Even And Odd Functions | Functions Math Help

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Even and Odd Functions

By Tina Goosz

With even and odd functions, we are determining whether a function is even or odd. Even functions are symmetrical around the y-axis and odd functions are symmetrical around the origin. With odd functions, half the graph on the y-axis is the upside down version of the other half of the y-axis so it is symmetrical around the origin - this is a hallmark indicator of odd functions.

Why is this concept useful?: This concept is useful to help us determine the symmetry of a function which is a practice application of math. This also provides us a connection between geometry and algebra being able to discuss the symmetry of the function graphically and have an algebraic link with the function. Also, this knowledge will provide a link to integrals in calculus.
Where does this concept fit into the curriculum?: High School Math

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How can we use the concept:

To determine if a function is even or odd, plug -x in the function for x and then simplify. If you ended up with f(-x) = f(x) where all the signs are the same, then the function is even. If you end up with the exact opposite, so f(-x) = -f(x) where all the signs are switched then the function is odd. In any other cases, the function is neither even nor odd.

Sample Math Problems

1. Determine algebraically whether

f(x) = -6x^{2} +5

is even, odd, or neither.

Solution:

\begin{array}{l} Plug\ -x\ in\ for\ x.\\ f( -x) \ =-6( -x)^{2} \ +5\\ \ f( -x) \ =\ -6x^{2} +\ 5\\ f( x) \ =\ -6x^{2} +\ 5\ \\ and\ f( -x) =-6x^{2} \ +\ 5\ \\ These\ are\ the\ same\ so\ the\ function\ is\ even.\ \end{array}

2. Determine algebraically whether

f(x) = 12x^{3} -4x

is even, odd, or neither.

Solution:

\begin{array}{l} Plug\ -x\ in\ for\ x.\\ f( -x) =\ 12( -x)^{3} \ -\ 4( -x)\\ f( -x) =\ -12x^{3} +\ 4x\\ In\ this\ case\ the\ f( -x) \ has\ the\ opposite\ signs\ as\ f( x) \ so\ this\ function\ is\ odd. \end{array}

3. Determine algebraically whether

f(x) = 7x^{3} +2x^{2} -3x+5

is even, odd, or neither.

4. Determine algebraically whether

f(x) = x^{4} +2x^{2} +4

is even, odd, or neither.

Solution:

\begin{array}{l} Plug\ -x\ in\ for\ x.\\ f( -x) \ =( -x)^{4} \ +\ 2( -x)^{2} +\ 4\\ f( -x) =\ x^{4} \ +\ 2x^{2} \ +\ 4\\ In\ this\ case\ f( -x) \ =\ f( x) \ so\ the\ function\ is\ even \end{array}

Functions Worksheets

Even and Odd Functions

Related Topics

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