Why are even functions called ‘even’, and odd functions called ‘odd’?

A function y = f(x) is an

even function of x if f(-x) = f(x)

odd function of x if f(-x) = -f(x)

for every x in the function’s domain.

The names even and odd comes from the powers of \boldsymbol{x}.

Even function:

If y is an even power of x as in y = x^{2} and y = x^{4} , it is an even function of x because

(-x)^{2} = x^{2}

(-x)^{4} = x^{4}

Odd function:

If y is an odd power of x as in y = x and y = x^{3} , it is an odd function of x because

(-x)^{1} = -x

(-x)^{3} = -x^{3}

Note that an even exponent does not necessarily implies that the function is even.

Example, f(x) = (x + 1)^2

In this case f(-x) = (-x + 1)^2 \neq f(x)  

Hence, f(x) = (x + 1)^2 is not an even function.

This is because on expanding (x + 1)^2 , we see that f(x) contains an odd power of x as well.

f(x) = (x + 1)^2 = x^2 + 2x + 1

Similarly, an odd exponent does not necessarily implies that the function is odd.

Example, f(x) = (x + 1)^3

In this case f(-x) = (-x + 1)^3 \neq -f(x)  

Hence, f(x) = (x + 1)^3 is not an odd function.

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Why are even functions called ‘even’, and odd functions called ‘odd’?

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