Power functions, even and odd functions

A power function is a function of the form $\displaystyle {y}={x^n}$, where $\displaystyle n$ is a natural number: $\displaystyle {y}={x}$, $\displaystyle {y}={x^2}$, $\displaystyle {y}={x^3}$, $\displaystyle {y}={x^4}$ and so on.

• They are all defined for every $\displaystyle x$.
• All their graphs pass through the points $\displaystyle (0,0)$ and $\displaystyle (1,1)$.
• The shape of the graph depends mainly on whether $\displaystyle n$ is even or odd.

When $\displaystyle n$ is even ($\displaystyle x^2$, $\displaystyle x^4$, $\displaystyle x^6$ …), the graph looks like a parabola:

• $\displaystyle {(-x)^n}={x^n}$, so the graph is symmetric about the $\displaystyle y$-axis and also passes through $\displaystyle (-1,1)$.
• $\displaystyle x^n\geq0$ for every $\displaystyle x$: the graph is never below the $\displaystyle x$-axis.
• $\displaystyle (0,0)$ is a minimum point: the function falls where $\displaystyle x<0$ and rises where $\displaystyle x>0$.
• As $\displaystyle x$ grows without end, and also as $\displaystyle x$ falls without end, $\displaystyle y$ grows without end: both ends of the graph point upwards.

The board below shows $\displaystyle {y}={x^2}$ and $\displaystyle {y}={x^4}$: near the origin $\displaystyle x^4$ is flatter, and outside $\displaystyle -1<x<1$ it is steeper.

When $\displaystyle n$ is odd ($\displaystyle x$, $\displaystyle x^3$, $\displaystyle x^5$ …):

• $\displaystyle {(-x)^n}={-x^n}$, so the graph is symmetric about the origin and also passes through $\displaystyle (-1,-1)$.
• The function is negative where $\displaystyle x<0$ and positive where $\displaystyle x>0$.
• The function rises everywhere, and has no extremum points.
• As $\displaystyle x$ grows without end, $\displaystyle y$ grows without end; as $\displaystyle x$ falls without end, $\displaystyle y$ falls without end too: the left end of the graph points down, the right end up.

When $\displaystyle x>1$, a higher power grows faster. For example, when $\displaystyle {x}={10}$:

Between 0 and 1 it is the other way round: when $\displaystyle {x}={0.5}$, $\displaystyle {x^2}={0.25}$ and $\displaystyle {x^4}={0.0625}$ – the higher power is smaller.

So when $\displaystyle x$ is very large, the highest power decides a polynomial like $\displaystyle x^3-100x$: when $\displaystyle {x}={1000}$, $\displaystyle x^3$ is a thousand million, and $\displaystyle 100x$ only a hundred thousand.

At the ends of the graph of a polynomial, as $\displaystyle x$ grows or falls without end, the term with the highest power decides where the graph goes.

• Odd degree: one end of the graph points down and the other up. The graph is continuous – it has no jumps – so it crosses the $\displaystyle x$-axis at least once: a polynomial of odd degree has at least one zero.
• Even degree: both ends point the same way, so the graph may never reach the $\displaystyle x$-axis. For example, $\displaystyle x^4+1\geq1$ for every $\displaystyle x$, and it has no zeros.

The example below finds between which whole numbers $\displaystyle {f(x)}={x^3-2x-5}$ has a zero:

• $\displaystyle {f(2)}={8-4-5}={-1}$, negative.
• $\displaystyle {f(3)}={27-6-5}={16}$, positive.
• The graph goes from below the $\displaystyle x$-axis to above it, so there is a zero between 2 and 3.

The properties of $\displaystyle x^n$ give names to any function:

• An even function: $\displaystyle {f(-x)}={f(x)}$ for every $\displaystyle x$ in the domain. The graph is symmetric about the $\displaystyle y$-axis.
• An odd function: $\displaystyle {f(-x)}={-f(x)}$ for every $\displaystyle x$ in the domain. The graph is symmetric about the origin.
• A polynomial with only even powers (and a constant) is even. A polynomial with only odd powers is odd.

The example below checks three functions:

• $\displaystyle {f(x)}={x^4-3x^2}$: $\displaystyle {f(-x)}={(-x)^4-3(-x)^2}={x^4-3x^2}={f(x)}$ – even.
• $\displaystyle {g(x)}={x^3-2x}$: $\displaystyle {g(-x)}={-x^3+2x}={-g(x)}$ – odd.
• $\displaystyle {h(x)}={x^2+x}$: $\displaystyle {h(1)}={2}$ and $\displaystyle {h(-1)}={0}$, so $\displaystyle h(-1)$ is neither $\displaystyle h(1)$ nor $\displaystyle -h(1)$ – neither even nor odd.

The functions are $\displaystyle {f(x)}={x^2}$ and $\displaystyle {g(x)}={x^4}$.

Mark on the board all the points where the two graphs meet.

Work out the $\displaystyle x$-coordinates of the meeting points.

Solve the inequality $\displaystyle x^4<x^2$: where is the graph of $\displaystyle g$ below the graph of $\displaystyle f$?

The function is $\displaystyle {f(x)}={2x^5-x^3}$.

Work out $\displaystyle f(-x)$.

Is $\displaystyle f$ even, odd or neither?
Explain.

Decide for each function whether it is even, odd or neither.
Explain each answer.

• $\displaystyle {f(x)}={x^6-x^2+7}$
• $\displaystyle {g(x)}={x^5+4x}$
• $\displaystyle {h(x)}={x^3+1}$
• $\displaystyle {k(x)}={(x-1)^2}$

The function is $\displaystyle {f(x)}={x^3+x+1}$.

Work out $\displaystyle f(-1)$ and $\displaystyle f(0)$.

Explain why $\displaystyle f$ has a zero between $\displaystyle -1$ and 0.
Explain why every polynomial of degree three has at least one zero.

$\displaystyle f$ is an even function, and $\displaystyle {f(2)}={7}$.
$\displaystyle g$ is an odd function defined for every $\displaystyle x$, and $\displaystyle {g(3)}={-4}$.
Write $\displaystyle f(-2)$ and $\displaystyle g(-3)$.
Explain why $\displaystyle {g(0)}={0}$.