Question

How do you tell whether a function is odd, even or neither?

Answer 1

Hint: This type of question is based on the concept of solving function. Let us consider f(x) to be a function of x. Now, we have to substitute -x in the function, that is, f(-x). Obtain the value of f(-x). Based on the result, we can find whether the function is odd, even or neither.

Complete step-by-step answer:
According to the question, we are asked to tell how to determine a function is odd, even or neither.
Let us assume f(x) to be a function of x.
Now, to determine whether a function is odd, even or neither odd nor even, we have to first substitute -x in the assumed function.
Therefore, consider f(-x).
If f(-x) is equal to f(x), we can say that the function f(x) is even.
That is, in even functions f(-x)=f(x).
We will understand more with an example.
Let us consider f(x)=|x|=x.
Therefore, f(-x)=|-x|
We know that |-x|=x and |x|=x. Using this property, we get
f(-x)=x=f(x)
From the above, we get f(x)=f(-x).
Therefore, that function f(x)=|x| is an even function.
Now, let us look into the odd function.
When f(-x) is equal to the negative of f(x), then the function is an odd function.
That is, in odd functions f(-x)=-f(x).
Let us consider an example to understand an example to understand more.
let f(x)=x.
Therefore, f(-x)=-x
But we know that f(x)=x
On further simplifications, we get
f(-x)=-(x)=-f(x)
Therefore, f(-x)=-f(x).
Now let us look into neither odd nor even functions.
When f(-x) is not equal to either f(x) or f(-x), we say the function is neither odd nor even.
That is, f(-x)≠f(x) and f(-x)≠-f(x).
Let us consider an example for more understanding.
Consider f(x)=2x+1.
Therefore, f(-x)=2(-x)+1.
on further simplification, we get
f(-x)=-2x+1
Here, f(-x)≠f(x) and f(-x)≠-f(x).
Therefore, the considered function f(x)=2x+1 is neither odd nor even.
Hence, the conditions for odd, even and neither odd nor even functions are known.

Note: Whenever you get this type of problem, we should always try to find the value of f(-x) which is the only way to solve. We should also avoid calculation mistakes based on sign conventions. Also don’t get confused with odd and even functions.