Odd functions

Let us take the odd function $f(x)=x^3$ and plot it on a set of axes.

$f(x)=x^3$ - StudySmarter Originals

If we were to rotate the graph 180${}^{\circ }$ about the origin, the result would look the same as the original graph. We can see thus say that this function is symmetrical about the origin.

We are given the function $f(x)=-7x^3+12x$.

Determine if this function is odd using algebraic methods.

Solution

Step 1: First determine $f(-x)$ by substituting $-x$ in the place of $x$ in the function and simplifying.

$$\begin{array}{rl}f(-x)& =-7(-x{)}^3+12(-x)\\ & =7x^3-12x\end{array}$$

Step 2: Next determine $-f(x)$.

$$\begin{array}{rl}-f(x)& =-(-7x^3+12x)\\ & =7x^3-12x\end{array}$$

$f(-x)=-f(x)$ for all values of $x$ in the domain of $f(x)$, we can therefore conclude that $f(x)$ is an odd function.

Let us use the same function from the previous example, $f(x)=-7x^3+12x$.

The graph of the function looks like this:

$f(x)=-7x^3+12x$ - StudySmarter Originals

If we were to rotate the function ${180}^{\circ }$ around the origin (clockwise or anticlockwise, the direction doesn't matter), the resulting graph would look exactly the same as the original.

We know that odd functions are symmetrical about the origin (i.e. rotating them ${180}^{\circ }$ around $(0,0)$) results in a graph identical to the original graph.

An easy way of doing this would be drawing a rough sketch of the graph on a scrap piece of paper and twisting the paper around ${180}^{\circ }$ to see if it looks the same.

Let us consider the function, $y=x^3$.

For all values of $x$, the value of $f(-x)$ will always be equal to $-f(x)$.

The graph of the function will be as follows:

Graph of $y=x^3$ - StudySmarter Originals

As we can see, if one were to rotate the graph ${180}^{\circ }$ around the origin (point (0,0)) then the resultant figure would look identical to the original. We can thus say that $x^3$ is symmetrical about the origin.

Another group of functions that often yield odd functions are trigonometric functions. For example, consider $y=sin(x)$.

We have $f(-x)=sin(-x)$ and $-f(x)=-sin(x)$. We know from trigonometry that $sin(-x)$ is the same as saying $-sin(x)$ and so $f(-x)$ therefore equals $-f(x)$ for all values of $x$ in the domain of the function. Thus, $y=sin(x)$ is an odd function.

If we look at the graph for $y=sin(x)$, we can also see how it is symmetrical about the origin:

Graph of $y=sin(x)$ - StudySmarter Originals

Other odd trigonometric functions include $tan(x),cot(x)$ and $cosec(x)$. These all satisfy the definition of an odd function.

You are given the function $g(x)=\frac{cos(x)}{x}$. Determine if the function is an odd function by using algebraic methods.

Solution

Step 1: Determine $g(-x)$.

$$\begin{array}{rl}g(-x)& =\frac{cos(-x)}{-x}\\ & =\frac{cos(x)}{-x}\\ & =-\frac{cos(x)}{x}\end{array}$$

Step 2: Determine $-g(x)$.

$$-g(x)=-\frac{cos(x)}{x}$$

Step 3: Is $g(x)$ an odd function?

Yes. $g(-x)=-g(x)$ for all values of $x$ in the domain of the function so $g(x)$ is therefore an odd function.