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# Function symmetry introduction

Learn what even and odd functions are, and how to recognize them in graphs.

#### What you will learn in this lesson

A shape has reflective symmetry if it remains unchanged after a reflection across a line.
For example, the pentagon above has reflective symmetry.
Notice how line $l$ is a line of symmetry, and that the shape is a mirror image of itself across this line.
This idea of reflective symmetry can be applied to the shapes of graphs. Let's take a look.

## Even functions

A function is said to be an even function if its graph is symmetric with respect to the $y$-axis.
For example, the function $f$ graphed below is an even function.
Verify this for yourself by dragging the point on the $x$-axis from right to left. Notice that the graph remains unchanged after a reflection across the $y$-axis!

1) Which of the graphs represent even functions?

### An algebraic definition

Algebraically, a function $f$ is even if $f\left(-x\right)=f\left(x\right)$ for all possible $x$ values.
For example, for the even function below, notice how the $y$-axis symmetry ensures that $f\left(x\right)=f\left(-x\right)$ for all $x$.

## Odd functions

A function is said to be an odd function if its graph is symmetric with respect to the origin.
Visually, this means that you can rotate the figure ${180}^{\circ }$ about the origin, and it remains unchanged.
Another way to visualize origin symmetry is to imagine a reflection about the $x$-axis, followed by a reflection across the $y$-axis. If this leaves the graph of the function unchanged, the graph is symmetric with respect to the origin.
For example, the function $g$ graphed below is an odd function.
Verify this for yourself by dragging the point on the $y$-axis from top to bottom (to reflect the function over the $x$-axis), and the point on the $x$-axis from right to left (to reflect the function over the $y$-axis). Notice that this is the original function!

Which of the graphs represent odd functions?

### An algebraic definition

Algebraically, a function $f$ is odd if $f\left(-x\right)=-f\left(x\right)$ for all possible $x$ values.
For example, for the odd function below, notice how the function's symmetry ensures that $f\left(-x\right)$ is always the opposite of $f\left(x\right)$.

## Reflection question

Can a function be neither even nor odd?

## Want to join the conversation?

• What is the use of describing a function as "even" or "odd"?
• Even and odd functions have properties that can be useful in different contexts. The most basic one is that for an even function, if you know f(x), you know f(-x). Similarly for odd functions, if you know g(x), you know -g(x). Put more plainly, the functions have a symmetry that allows you to find any negative value if you know the positive value, or vice versa.
• Can an equation be both even and odd?
• The only function which is both even and odd is f(x) = 0, defined for all real numbers. This is just a line which sits on the x-axis. If you count equations which are not a function in terms of y, then x=0 would also be both even and odd, and is just a line on the y-axis.
• How can you prove definitively that a function is even or odd (or neither) just by its equation? Is there even a way?
• Mona's explanation works very well for polynomials. Two things to keep in mind:

1) Odd functions cannot have a constant term because then the symmetry wouldn't be based on the origin.

2) Functions that are not polynomials or that don't have exponents can still be even or odd. For example, f(x)=cos(x) is an even function.
• How can a function be neither even or odd?
• Even and odd describe 2 types of symmetry that a function might exhibit.
1) Functions do not have to be symmetrical. So, they would not be even or odd.
2) If a function is even, it has symmetry around the y-axis. What is a function has symmetry around y=5? It would not be even, because the symmetry is not around the Y-axis.
3) Similarly, odd functions have symmetry around the origin. Functions might have symmetry based on some point other than the origin. So, they would not be odd.
Hope this helps.
• Let's say the parent function y=x^2 gets translated to the left by 4. So now the equation is y=(x+4)^2. Is it still an even function? It is confusing because now the graph is not symmetric over the y-axis. So does this mean it is an odd function now? Or is it neither?
• Even function are strictly symmetrical about the y axis, so it's neither.
• In the first question, it asks to select all the even functions. Wouldn't the final option not be correct because it isn't a function?
(1 vote)
• Why do you think it isn't a function? Remember, a function is just a set of ordered pairs where each input (x-value) has exactly one output (y-value). The final graph meets that criteria. Functions do not need to a continuous line. They don't even need to be lines, they can just be a collections of points.
• Can a function be neither even nor odd? i think the answer is yes im not quite sure
• Yes. There are functions which are neither even nor odd, as they don't exhibit any sort of symmetry. One simple example is f(x) = e^x.
• How about symmetry with respect to x-axis only? Is it a thing?
Why did we define an even function to be symmetric with respect to y-axis and not the x one?
• Remember the vertical line test? A curve cannot be a function when a vertical line interesects it more than once.

And a curve that is symmetrical around the x-axis will always fail the vertical line test (unless that function is f(x) = 0). So, a function can never be symmetrical around the x-axis.

Just remember:
symmetry around x-axisfunction

To answer your second question, "even" and "odd" functions are named for the exponent in this power function:

f(x) = xⁿ

- if n is an even integer, then f(x) is an "even" function
- if n is an odd integer, then f(x) is an "odd" function

Hope this helps!