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Math Lesson 16.7.6 - The Graph of Odd Functions
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Welcome to our Math lesson on The Graph of Odd Functions, this is the sixth lesson of our suite of math lessons covering the topic of Even and Odd Functions, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
The Graph of Odd Functions
Since in odd functions f(-x) = -f(x) for any x, if for example f(x) is located at the upper-right side of the coordinates system (first quadrant), then f(-x) lies on the upper left side of the coordinates system (second quadrant). On the other hand, -f(x) lies on the bottom-right side (fourth quadrant), so f(-x) and -f(x) lie on the opposite parts of the coordinates system, as shown in the figure below, where the function f(x) = x3 is used for illustration.
It is easy to see that making half a rotation of f(-x) around the origin gives -f(x). This is a feature that distinguishes odd functions from non-odd ones.
Example 6
Which of the functions shown in the graphs below is odd?
Solution 6
- From the graph, it is evident that the function f(x) is odd, as rotating the right part of the graph by half a cycle gives the left part of the same graph and vice-versa.
Indeed, using known techniques to identify the function's formula when its graph is given results in the function shown in the figure is f(x) = -x/3, which is proven to be odd. - From the graph, it is evident that the function g(x) is not odd (it is even indeed), as rotating the right part of the graph by half a cycle does not give the left part of the same graph and vice-versa.
Indeed, using the known techniques to identify the function's formula when its graph is given, provides results that the function shown in the figure is g(x) = x2 - 2, which is proven to be even. - From the graph, it is evident that the function h(x) is odd, as rotating the right part of the graph by half a cycle gives the left part of the same graph and vice-versa.
Indeed, using the known techniques to identify the function's formula when its graph is given, provides results that the function shown in the figure is h(x) = 4x - x3, which is proven to be odd.
You have reached the end of Math lesson 16.7.6 The Graph of Odd Functions. There are 10 lessons in this physics tutorial covering Even and Odd Functions, you can access all the lessons from this tutorial below.
More Even and Odd Functions Lessons and Learning Resources
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- Continuing learning functions - read our next math tutorial: Relation and Function
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