If a function is odd its graph is symmetric with respect to the origin. Explain.
Functions are very important concepts in mathematics. Every advanced topic in mathematics has to deal with functions at some point in time; be it trigonometry, linear algebra, or calculus. We can tell if a function is even or not by looking at its graph. Let's see how.
Answer: The graph of an odd function which is always symmetric with respect to the origin, satisfies the condition f(-x) = -f(x).
Let's understand in detail.
Let's understand this with an example.
Let's prove the function f(x) = x3 odd using its graph.
First, we plot the graph of f(x) = x3 as shown below. You can see that the graph is symmetric about the origin.
Then, we plot for f(-x) = -x3. This is nothing but inverting the graph about the y-axis.
If we look closely, then we see that if we invert the graph with respect to the x-axis, then we get f(x) back.
This inverting about the x-axis is equivalent to multiplying the graph by -1.
Hence, we prove f(x) = -f(-x) for f(x) = x3. Hence it is an odd function.
Hence, by this example, we understand that the graph of an odd function that is always symmetric with respect to the origin, satisfies the condition f(-x) = -f(x).