##### 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’?