The properties of a function: even and odd functions

Algebra and calculus, grade 10 (4 units) Lesson 4 of 20Learning

The properties of a function: even and odd functions

The properties of a function

Investigating a function is describing its properties – what can be known about its graph even without drawing it:

the domain – the $\displaystyle x$ -values that can be put into the function;
the points where the graph meets the axes – the $\displaystyle x$ -axis where $\displaystyle {f(x)}={0}$ , and the $\displaystyle y$ -axis where $\displaystyle {x}={0}$ ;
where the function is positive – $\displaystyle f(x)>0$ , the graph above the $\displaystyle x$ -axis; where it is negative – $\displaystyle f(x)<0$ , the graph below it;
where it increases and where it decreases – where the graph goes up from left to right, and where it goes down;
extremum points – a point where the function turns from increasing to decreasing is a maximum point, and a point where it turns from decreasing to increasing is a minimum point.

The linear function

The graph of $\displaystyle {f(x)}={mx+b}$ is a line. The slope $\displaystyle m$ decides all of its increasing and decreasing:

$\displaystyle m>0$ – the function increases everywhere; $\displaystyle m<0$ – it decreases everywhere;
a linear function (one that is not constant) has no extremum point.

$\displaystyle {f(x)}={-2x+6}$ , for example:

The domain: every $\displaystyle x$ .
It meets the $\displaystyle y$ -axis at $\displaystyle (0,6)$ ; the $\displaystyle x$ -axis where $\displaystyle {-2x+6}={0}$ , that is at $\displaystyle (3,0)$ .
$\displaystyle {m}={-2<0}$ : the function decreases everywhere, so it is positive to the left of 3 and negative to the right of it.

The quadratic function

The graph of $\displaystyle {f(x)}={ax^2+bx+c}$ is a parabola, and it has one extremum point – the vertex. Its $\displaystyle x$ -coordinate:

\displaystyle x

\displaystyle =

\displaystyle \frac{-b}{2a}

The example below investigates $\displaystyle {f(x)}={x^2-2x-3}$ :

\displaystyle f(x)=x^{2}-2x-3

\displaystyle (-1,0)

\displaystyle (3,0)

\displaystyle (0,-3)

\displaystyle (1,-4)

\displaystyle (2,-3)

The domain: every $\displaystyle x$ .
The $\displaystyle x$ -axis: $\displaystyle {x^2-2x-3}={0}$ gives $\displaystyle (-1,0)$ and $\displaystyle (3,0)$ ; the $\displaystyle y$ -axis: $\displaystyle (0,-3)$ .
The vertex: $\displaystyle {x}={\frac{2}{2}}={1}$ , $\displaystyle {f(1)}={-4}$ . The parabola opens upwards ( $\displaystyle {a}={1>0}$ ), so $\displaystyle (1,-4)$ is a minimum point.
The function decreases where $\displaystyle x<1$ and increases where $\displaystyle x>1$ .
Positive where $\displaystyle x<-1$ or $\displaystyle x>3$ ; negative where $\displaystyle -1<x<3$ .
The parabola is symmetric about the line $\displaystyle {x}={1}$ : the points $\displaystyle (0,-3)$ and $\displaystyle (2,-3)$ are at the same height, because they are the same distance from 1.

Even and odd functions

An even function gives the same value at $\displaystyle x$ and at $\displaystyle -x$ ; an odd function gives opposite values:

\displaystyle f(-x)

\displaystyle =

\displaystyle f(x)

\displaystyle f(-x)

\displaystyle =

\displaystyle -f(x)

On the board – $\displaystyle {f(x)}={x^4-3x^2}$ , even, and $\displaystyle {g(x)}={x^3-4x}$ (the dashed one), odd:

\displaystyle f(x)=x^{4}-3x^{2}

\displaystyle g(x)=x^{3}-4x

The graph of an even function is symmetric about the $\displaystyle y$ -axis: the point $\displaystyle (x,y)$ is on it, and so is $\displaystyle (-x,y)$ .
The graph of an odd function is symmetric about the origin: the point $\displaystyle (x,y)$ is on it, and so is $\displaystyle (-x,-y)$ .
An odd function defined at 0 passes through $\displaystyle (0,0)$ , because $\displaystyle {f(0)}={-f(0)}$ .
Most functions are neither even nor odd.

Even or odd by algebra

To check whether a function is even, put $\displaystyle -x$ in place of $\displaystyle x$ and compare with $\displaystyle f(x)$ and with $\displaystyle -f(x)$ .

$\displaystyle {f(x)}={x^4-3x^2}$ : $\displaystyle {f(-x)}={(-x)^4-3(-x)^2}={x^4-3x^2}={f(x)}$ . Even.
$\displaystyle {g(x)}={x^3-4x}$ : $\displaystyle {g(-x)}={(-x)^3-4(-x)}={-x^3+4x}={-g(x)}$ . Odd.
$\displaystyle {h(x)}={x^2+x}$ : $\displaystyle {h(-x)}={x^2-x}$ , which is neither $\displaystyle h(x)$ nor $\displaystyle -h(x)$ . Neither even nor odd. One example is enough to show it: $\displaystyle {h(1)}={2}$ , but $\displaystyle {h(-1)}={0}$ .

An even power of $\displaystyle -x$ equals the power of $\displaystyle x$ , and an odd power is its opposite. So a polynomial with only even powers (and a number) is even, and a polynomial with only odd powers is odd.

Exercise 1

$\displaystyle {f(x)}={-x^2+4x+5}$

Mark on the board the points where the graph meets the axes, and the extremum point.

\displaystyle f(x)=-x^{2}+4x+5

Find where the function is positive.

What kind of extremum point is it?
Write where the function increases and where it decreases.

Exercise 2

$\displaystyle {f(x)}={-2x+6}$

Find where the function is positive.

Without calculating: why does the function have no extremum point?

Exercise 3

Decide whether each function is even, odd or neither. Give a reason.
$\displaystyle {f(x)}={2x^4-x^2}$
$\displaystyle {g(x)}={x^3+x}$
$\displaystyle {h(x)}={x^2-2x}$
$\displaystyle {k(x)}={x^5-x^3+1}$

Exercise 4

The table below belongs to an odd function $\displaystyle f(x)$ .
Complete it.

$\displaystyle x$	$\displaystyle -3$	$\displaystyle -1$	$\displaystyle 0$	$\displaystyle 1$	$\displaystyle 3$
$\displaystyle f(x)$	$\displaystyle 1$	$\displaystyle -2$

\displaystyle f(0)

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

\displaystyle f(1)

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

\displaystyle f(3)

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 5

The graph of an even function passes through the points $\displaystyle (2,5)$ and $\displaystyle (-3,1)$ .
Which two more points must be on the graph?
And if the function is odd?

Exercise 6

$\displaystyle {f(x)}={x^2-4}$

Mark on the board the points where the graph meets the $\displaystyle x$ -axis, and the extremum point.

\displaystyle f(x)=x^{2}-4

Show by algebra that the function is even.
How does being even show in the points you marked?

Find where the function is positive.

Formulas: algebra and calculus, grade 10 (4 units)

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Algebraic technique

The square of a sum
$\displaystyle (a+b)^2$ $\displaystyle =$ $\displaystyle a^2+2ab+b^2$
The square of a difference
$\displaystyle (a-b)^2$ $\displaystyle =$ $\displaystyle a^2-2ab+b^2$
The difference of squares
$\displaystyle (a+b)(a-b)$ $\displaystyle =$ $\displaystyle a^2-b^2$
Factoring a trinomial – $\displaystyle x_1$ , $\displaystyle x_2$ the roots of $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle ax^2+bx+c$ $\displaystyle =$ $\displaystyle a(x-x_1)(x-x_2)$
The roots of $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
The product of roots
$\displaystyle \sqrt{a}\cdot\sqrt{b}$ $\displaystyle =$ $\displaystyle \sqrt{ab}$ $\displaystyle a\geq0,\ b\geq0$
The quotient of roots
$\displaystyle \frac{\sqrt{a}}{\sqrt{b}}$ $\displaystyle =$ $\displaystyle \sqrt{\frac{a}{b}}$ $\displaystyle a\geq0,\ b>0$

The properties of a function

An even function – its graph symmetric about the $\displaystyle y$ -axis
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle f(x)$
An odd function – its graph symmetric about the origin
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle -f(x)$
The $\displaystyle x$ -coordinate of the vertex of $\displaystyle {y}={ax^2+bx+c}$
$\displaystyle x=\frac{-b}{2a}$
The absolute value
$\displaystyle |a|=\begin{cases}a & a\geq0\\-a & a<0\end{cases}$

Transforming a graph

A shift of $\displaystyle k$ units up ( $\displaystyle k<0$ – down)
$\displaystyle y=f(x)+k$
A shift of $\displaystyle p$ units to the right ( $\displaystyle p<0$ – to the left)
$\displaystyle y=f(x-p)$
A reflection in the $\displaystyle x$ -axis
$\displaystyle y=-f(x)$
A reflection in the $\displaystyle y$ -axis
$\displaystyle y=f(-x)$
A stretch ( $\displaystyle |a|>1$ ) or a shrink ( $\displaystyle |a|<1$ )
$\displaystyle y=a\cdot f(x)$

The derivative and the rules of differentiation

The average rate of change from $\displaystyle x_1$ to $\displaystyle x_2$ – the slope of the chord
$\displaystyle \frac{f(x_2)-f(x_1)}{x_2-x_1}$
The derivative at a point – the slope of the tangent
$\displaystyle f'(x_0)=\lim_{h\to0}\frac{f(x_0+h)-f(x_0)}{h}$
The derivative of a power
$\displaystyle (x^n)'$ $\displaystyle =$ $\displaystyle nx^{n-1}$
The derivative of a constant
$\displaystyle (c)'$ $\displaystyle =$ $\displaystyle 0$
The derivative of a constant multiple
$\displaystyle (c\cdot f(x))'$ $\displaystyle =$ $\displaystyle c\cdot f'(x)$
The derivative of a sum and of a difference
$\displaystyle (f(x)\pm g(x))'$ $\displaystyle =$ $\displaystyle f'(x)\pm g'(x)$
The derivative of a product
$\displaystyle (f(x)\cdot g(x))'$ $\displaystyle =$ $\displaystyle f'(x)\cdot g(x)+f(x)\cdot g'(x)$
The derivative of a composite function
$\displaystyle (f(g(x)))'$ $\displaystyle =$ $\displaystyle f'(g(x))\cdot g'(x)$

The tangent to a graph

The slope of the tangent at the point $\displaystyle (x_1,y_1)$ on the graph
$\displaystyle m=f'(x_1)$
The equation of the tangent
$\displaystyle y-y_1=m(x-x_1)$

The derivative of a square root function

The derivative of $\displaystyle \sqrt{x}$
$\displaystyle (\sqrt{x})'$ $\displaystyle =$ $\displaystyle \frac{1}{2\sqrt{x}}$ $\displaystyle x>0$
The derivative of the root of a function
$\displaystyle (\sqrt{f(x)})'$ $\displaystyle =$ $\displaystyle \frac{f'(x)}{2\sqrt{f(x)}}$ $\displaystyle f(x)>0$

Rising, falling and extremum points

The function rises
$\displaystyle f'(x)>0$
The function falls
$\displaystyle f'(x)<0$
A point that may be an extremum
$\displaystyle f'(x)=0$