Even, odd, increasing and decreasing functions

Symmetry

Look at your face in a mirror. The left side matches the right. Some graphs have this kind of symmetry too, and it saves you work: you draw one half, and the other half comes free.

A function is even if

The graph of an even function is symmetric about the $y$ axis. Examples: $x^2$, $x^4$, $|x|$ and $x^4 - 2x^2$.

A function is odd if

The graph of an odd function is symmetric about the origin: if you turn it half a turn about the origin, it falls on itself. Examples: $x$, $x^3$, $\dfrac{1}{x}$ and $x^3 - 2x$.

Many functions are neither, such as $x^2 + x$ in the third picture.

How to test a function

For the test, the domain must be symmetric, which means that if $x$ is in the domain, then $-x$ is also in it. Then find $f(-x)$ and compare it with $f(x)$.

Example. $f(x) = x^3 - 2x$. Then $f(-x) = (-x)^3 - 2(-x) = -x^3 + 2x = -(x^3 - 2x) = -f(x)$. So $f$ is odd.

Example. $f(x) = x^2 + x$. Then $f(-x) = x^2 - x$. This is not $f(x)$, and it is not $-f(x)$. So $f$ is neither even nor odd. To prove it, one value is enough: $f(1) = 2$, $f(-1) = 0$, and $0$ is neither $2$ nor $-2$.

Useful rules. Even $+$ even is even. Odd $+$ odd is odd. Even $\times$ even is even. Odd $\times$ odd is even. Even $\times$ odd is odd. An odd function that is defined at $0$ has $f(0) = 0$. Only the zero function $f(x) = 0$ is both even and odd.

Every function splits. If the domain is symmetric, then

For example, $x^2 + x$ is the even function $x^2$ plus the odd function $x$.

Set $a = 0$ and $c = 0$: the two curves are mirror images through the origin, so $f$ is odd. Set $b = 0$: they lie on top of each other, so $f$ is even.

Increasing and decreasing

A function $f$ is increasing on an interval if, for any two points $x_1 < x_2$ in it, $f(x_1) < f(x_2)$. Going to the right, the graph goes up. It is decreasing if $x_1 < x_2$ gives $f(x_1) > f(x_2)$, and the graph goes down.

The graph of $y = x^3 - 3x$ is increasing for $x < -1$, decreasing for $-1 < x < 1$ and increasing again for $x > 1$. The points where the direction changes, here $(-1, 2)$ and $(1, -2)$, are the turning points. At a turning point, the function has a highest or lowest value in its neighbourhood.

Example. $f(x) = x^2 - 4x$. Completing the square, $f(x) = (x - 2)^2 - 4$. For $x < 2$, a larger $x$ makes $(x - 2)^2$ smaller, so $f$ decreases. For $x > 2$, it increases. The turning point is the vertex, $(2, -4)$.

Periodic functions

A function is periodic if its values repeat. It has period $T$ if $f(x + T) = f(x)$ for every $x$, with $T > 0$, and $T$ is the smallest such number. The fractional part $\{x\}$ has period $1$. The functions $\sin x$ and $\cos x$, which you will meet soon, have period $2\pi$. If $f$ has period $T$, then $f(kx)$ has period $\dfrac{T}{k}$. So $\{3x\}$ has period $\dfrac{1}{3}$.

Practise with new numbers

Try it in Python

Python cannot prove that a function is even, but it can test many values of $x$. This program tests five functions.

Remember these

• Even: $f(-x) = f(x)$, symmetric about the $y$ axis. Odd: $f(-x) = -f(x)$, symmetric about the origin.
• Test: write $f(-x)$ and compare. One bad value is enough to show that a function is neither.
• Increasing: the graph goes up to the right. Decreasing: the graph goes down.
• $f$ has period $T$ if $f(x + T) = f(x)$. The fractional part has period $1$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Is $f(x) = x^5 - 2x$ even, odd or neither?

2. Is $f(x) = x^4 - 3|x|$ even, odd or neither?

3. On which interval is $f(x) = x^2 + 8x$ decreasing?

4. Can a function be both even and odd?