Functions have two main types of symmetry. If the function is even, its graph will be symmetric about the *y*-axis. If the function is odd, its graph will be symmetric with respect to the origin. Here, we will look at a summary of these types of function symmetry. Then, we will see the tests that are used to know if a function is symmetric or not.

Also, we will look at various function symmetry examples to fully master the use of these symmetry tests.

## Summary of symmetry of functions

The symmetry of functions can be used to obtain its graph more easily since that by knowing a portion of the graph, we will also know the other symmetric portion.

Symmetry depends on the behavior of $latex f(x)$ on the other side of the *y*-axis, that is, on negative values of *x*: $latex f(-x)$.

A function is symmetric with respect to the *y*-axis if every time we have (*a, b*) on the graph of the function, we also have (-*a, b*). The following is a graph with symmetry about the *y*-axis:

A function is symmetric with respect to the origin if every time we have (*a, b*), we also have (-a, -b) on the graph. The following is a graph with symmetry with respect to the origin:

### Tests for symmetry

**Proof 1:** If $latex f(-x)=f(x)$, then the graph of $latex f(x)$ is symmetric about the ** y-axis**. A function symmetric about the

*y*-axis is called an even function.

** Proof 2:** If $latex f(-x)=-f(x)$, then the graph of $latex f(x)$ is symmetric about the

**origin**. A function symmetric about the origin axis is called an odd function.

## Symmetry of functions – Examples with answers

The following examples in symmetry of functions are solved using the symmetry tests indicated above. Each example has a detailed solution to understand the reasoning used.

**EXAMPLE 1**

Determine if the function $latex f(x)=2{{x}^2}+5$ is symmetric.

##### Solution

**Test 1:**We start by checking if the function is symmetric with respect to the *y *axis. That means we have to replace all *x *with –*x*:

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

$latex ={{x}^2}+5$

After simplifying, we got the exact same function as $latex f(x)$, which means that they are both equivalents. Thus, this equation is symmetric with respect to the *y*-axis.

**Test 2: **Now, we check for symmetry with respect to the origin. This requires that we replace both *x *with –*x *and $latex f(x)$ with $latex -f(-x)$.

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

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

In this case, the right side of the function is identical to the original but the left side is different since we have a minus sign in front.

**EXAMPLE 2**

Determine if the function $latex f(x)={{x}^3}+5x$ has some kind of symmetry.

##### Solution

**Test 1:** To determine if the function is symmetric with respect to the *y*-axis, we have to replace all *x* with –*x*:

$latex f(-x)={{(-x)}^3}+5(-x)$

$latex =-{{x}^3}-5x$

We see that we obtained an expression different from the original function on the right-hand side. This means that the function is not symmetric with respect to the *y-*axis.

**Test 2: **To check if the function is symmetric with respect to the origin, we have to change both to *x *with –*x *and $latex f(x)$ with $latex -f(-x)$.

$latex -f(-x)={{(-x)}^3}+5(-x)$

$latex -f(-x)=-{{x}^3}-5x$

$latex f(-x)={{x}^3}+5x$

We see that we got negative signs from both sides. By multiplying both sides by -1, we got the original function. This means that the function itself is symmetric with respect to the origin.

**EXAMPLE 3**

Has the function $latex f(x)={{x}^4}+2{{x}^3}+2x$ some kind of symmetry?

##### Solution

**Test 1:** We look for symmetry with respect to the *y-*axis by replacing all *x *with –*x*:

$$f(-x)={{(-x)}^4}+2{{(-x)}^3}+2(-x)$$

$latex ={{x}^4}-2{{x}^3}-2x$

After simplifying, we got a different expression than the original function, $latex f(x)$. Therefore, this equation has no symmetry with respect to the *y-*axis.

**Test 2: **We look for symmetry with respect to the origin by replacing both *x *with –*x *and $latex f(x)$ with $latex -f(-x)$.

$$-f(-x)={{(-x)}^4}+2{{(-x)}^3}+2(-x)$$

$latex -f(-x)={{x}^4}-2{{x}^3}-2x$

$latex f(-x)=-{{x}^4}+2{{x}^3}+2x$

If we multiply both sides of the function by -1, we do not obtain the original function, so this function does not have symmetry with respect to the origin.

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**EXAMPLE 4**

Determine if the function $latex f(x)=3{{x}^4}-2{{x}^2}+4$ has some kind of symmetry.

##### Solution

**Test 1:** We start by looking for symmetry with respect to the *y-*axis. That means we have to replace all *x *with –*x*:

$latex f(-x)=3{{(-x)}^4}-2{{(-x)}^2}+4$

$latex =3{{x}^4}-2{{x}^2}+4$

We got exactly the same function as $latex f(x)$. Therefore, the function does have symmetry with respect to the *y-*axis.

**Test 2: **Now, we look for symmetry with respect to the origin. We need to replace both *x *with –*x *and $latex f(x)$ with $latex -f(-x)$.

$latex -f(-x)=3{{(-x)}^4}-2{{(-x)}^2}+4$

$latex -f(-x)=3{{x}^4}-2{{x}^2}+4$

$latex f(-x)=-3{{x}^4}+2{{x}^2}-4$

After multiplying by -1, we didn’t get the original function. This means that the function is not symmetric about the origin.

**EXAMPLE 5**

Is the function $latex f(x)={{x}^3}+{{x}^2}+x+1$ symmetric with respect to the *y*-axis or the origin?

##### Solution

**Test 1:** We check symmetry with respect to the *y*-axis by replacing all *x* with –*x*:

$$f(-x)={{(-x)}^3}+{{(-x)}^2}+(-x)+1$$

$latex =-{{x}^3}+{{x}^2}-x+5$

We see that we did not obtain the same function as the original function, $latex f(x)$, which means that the function is not symmetric with respect to the *y-*axis.

**Test 2: **We can check if the function has symmetry with respect to the original by replacing both *x *with –*x *and $latex f(x)$ with $latex -f(-x)$.

$$-f(-x)={{(-x)}^3}+{{(-x)}^2}+(-x)+1$$

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

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

After simplifying and multiplying the function by -1, we did not obtain the original function, which means that it is not symmetric with respect to the origin. This function does not have any kind of symmetry.

This is not unusual since most functions do not have any kind of symmetry.

**EXAMPLE 6**

Determine if the function $latex f(x)=3{{x}^5}-4{{x}^3}+2x+4$ has some kind of symmetry.

##### Solution

**Test 1:** We replace all *x *with –*x *to look for symmetry with respect to the *y*-axis*:*

$$f(-x)=3{{(-x)}^5}-4{{(-x)}^3}+2(-x)+4$$

$latex =-3{{x}^5}+4{{x}^3}-2x+4$

After simplifying, we did not get the original function, which means that both are not equivalent and there is no symmetry with respect to the *y-*axis.

**Test 2: **We replace both x with –x and $latex f(x)$ with $latex -f(-x)$ to find symmetry with respect to the origin:

$$-f(-x)=3{{(-x)}^5}-4{{(-x)}^3}+2(-x)+4$$

$$-f(-x)=-3{{x}^5}+4{{x}^3}-2x+4$$

$$f(-x)=3{{x}^5}-4{{x}^3}+2x-4$$

We didn’t exactly get the original function, so the function is not symmetric about the origin.

## Symmetry of functions – Practice problems

Practice and test your knowledge of function symmetry with the following exercises. Solve the problems and select an answer. You can look at the solved examples above in case you need help.

## See also

Interested in learning more about functions? Take a look at these pages:

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