If someone were to point and shout "Hey, look, it's an odd function!" you would likely think that they had seen a weirdly shaped graph. In actuality, odd functions have symmetrical graphs and the word 'odd' is used in its mathematical context instead of to mean 'something strange'.
One would also be likely to assume that odd functions are the exact opposite of even functions and call it a day. That assumption would be terrifically wrong. The difference between odd and even functions can't just be classified by thinking that if one's not even, then it must be odd.
As we delve deeper into the topic of odd functions, we'll see what formula odd functions use, as well as how that ties into the symmetry of such functions.
What are odd functions?
Functions can be classified as even, odd or neither depending on the symmetry of the function. An odd function is one that is symmetrical about the point \((0,0)\). Simply put, if you take the graph of an odd function and rotate it 180\(^{\circ}\) around the origin, \((0,0)\), on a set of axes, the resultant graph looks identical to the original one.
The formal definition of an odd function is as follows:
An odd function is a function for which \(f(-x)=-f(x)\) for all values of \(x\) in the domain of \(f\).
This means that for every \(x\)-value within the domain of an odd function \(f(x)\), the function value \(f(-x)\) is the same as the value \(-f(x)\).
This can also be shown graphically and is discussed in more detail in the next sections.
Graph of an odd function
We already know that odd functions are symmetrical about the origin and that they satisfy \(f(-x)=-f(x)\) for every value of \(x\) in their domains. By plotting the graph of an odd function, we can see the symmetry of the function.
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.
Difference between even and odd functions
It is recommended that you first cover the topic Even Functions before proceeding with this section!
The main difference between even and odd functions is their axes of symmetry. Odd functions, as we know, are symmetric about the origin. Even functions, on the other hand, are symmetric about the y-axis.
This means that the shape of any even function graph will be mirrored perfectly over the y-axis. A good example of an even function is \(x^{2}\). The shape of the parabola to the left of the y-axis is a mirror image of the shape to the right of the y-axis. This is shown in the image below.
Graph of the even function \(y = x^{2}\) - StudySmarter Originals
The following table summarises the two main differences between odd and even functions:
Odd Functions | Even Functions |
| \(f(-x) = -f(x)\) for all values of \(x\) in the domain of \(f(x)\) | \(f(-x) = f(x)\) for all values of \(x\) in the domain of \(f(x)\) |
| The graph of an odd function is symmetrical about the origin | The graph of an even function is symmetrical about the y-axis |
How to determine if a function is odd
We can use the differences mentioned in the previous section to determine either graphically or algebraically if a function is odd.
Determining if a function is even or odd algebraically
To determine if a function is odd using algebraic methods, one just has to apply the formula \(f(-x) = -f(x)\) and see if it holds true.
The following example demonstrates how to do this:
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.
\[\begin{equation}\begin{split}f(-x) & = -7(-x)^{3} + 12(-x) \\& = 7x^{3} -12x\end{split}\end{equation}\]
Step 2: Next determine \(-f(x)\).
\[\begin{equation}\begin{split}-f(x) & = -(-7x^{3} + 12x) \\& = 7x^{3} - 12x\end{split}\end{equation}\]
\(f(-x) = -f(x)\) for all values of \(x\) in the domain of \(f(x)\), we can therefore conclude that \(f(x)\) is an odd function.
Determining if a function is odd graphically
The easiest way of determining if a function is odd is by using its graph. Graphs provide us with a better way of visualizing functions and are useful for interpreting them as well.
The following example demonstrates how to determine if a function is odd graphically:
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.
It is possible for a function to be neither even nor odd, so worry not if your given function does not satisfy a definition for even or odd functions.
Odd functions examples
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.
Similarly, functions such as \(x, x^{5}, x ^{7}, x^{9}, x^{11}, x^{-1}, x^{-3},\) etc. are all odd functions. For these functions, \(f(-x) = -f(x)\) for all values of \(x\). That is to say, if you were to substitute \(-x\) into each of these functions and simplify then the resulting answer would the same as multiplying each of these functions by \(-1\).
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.
Odd functions can also be fractions. Consider the following example:
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 \(g(-x)\).
\[\begin{equation}\begin{split}g(-x) & = \frac{cos(-x)}{-x} \\& = \frac{cos(x)}{-x} \\& = - \frac{cos(x)}{x}\end{split}\end{equation}\]
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.
Odd functions - Key takeaways
- Odd functions are functions in which \(f(-x) = -f(x)\).
- Odd functions are symmetric about the origin. This means that if you were to rotate the graph of an odd function \(180^{\circ}\) around the origin point, the resulting graph would look identical to the original.
- One can determine if a function is odd by using algebraic or graphical methods.
- It is possible for a function to be neither even nor odd.
- Trigonometric functions can also be odd functions.
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