The absolute value, operations on functions and composition

Calculus, grade 10 (5 units) Lesson 6 of 26Learning

The absolute value, operations on functions and composition

The absolute value function

The absolute value of a number is its distance from 0 on the number line: $\displaystyle {|5|}={5}$ , $\displaystyle {|-5|}={5}$ , $\displaystyle {|0|}={0}$ .

\displaystyle |a|=\begin{cases}a & a\geq0\\-a & a<0\end{cases}

The graph of $\displaystyle {y}={|x|}$ is made of two straight pieces:

Where $\displaystyle x\geq0$ it is the line $\displaystyle {y}={x}$ , and where $\displaystyle x<0$ – the line $\displaystyle {y}={-x}$ .
$\displaystyle {|-x|}={|x|}$ : the function is even, and the graph is symmetric about the $\displaystyle y$ -axis.
$\displaystyle (0,0)$ is a minimum point, and the graph makes a sharp corner there.

\displaystyle y=|x|

The graph of

\displaystyle |f(x)|

The graph of $\displaystyle {y}={|f(x)|}$ comes from the graph of $\displaystyle f$ :

Where $\displaystyle f(x)\geq0$ , the graph does not change.
Where $\displaystyle f(x)<0$ , the part below the $\displaystyle x$ -axis is reflected above it.
The zeros stay where they are, and $\displaystyle |f(x)|$ is never negative.

The board below shows $\displaystyle {y}={x^2-4}$ (dashed) and $\displaystyle {y}={|x^2-4|}$ . The part between $\displaystyle -2$ and 2 went up above the axis, and the minimum point $\displaystyle (0,-4)$ became a maximum point $\displaystyle (0,4)$ .

\displaystyle y=x^{2}-4

\displaystyle y=|x^{2}-4|

Equations and inequalities with an absolute value

$\displaystyle {|A|}={c}$ (where $\displaystyle c>0$ ) says that the distance of $\displaystyle A$ from 0 is $\displaystyle c$ , so $\displaystyle {A}={c}$ or $\displaystyle {A}={-c}$ . $\displaystyle |A|<c$ says that $\displaystyle A$ is closer to 0 than $\displaystyle c$ : $\displaystyle -c<A<c$ .

The examples below solve an equation and an inequality:

$\displaystyle {|x-3|}={5}$ : $\displaystyle {x-3}={5}$ or $\displaystyle {x-3}={-5}$ , so $\displaystyle {x}={8}$ or $\displaystyle {x}={-2}$ .
$\displaystyle |2x+1|<5$ : $\displaystyle -5<2x+1<5$ , that is $\displaystyle -6<2x<4$ , so $\displaystyle -3<x<2$ .

Operations on functions

From two functions $\displaystyle f$ and $\displaystyle g$ , each of the four operations of arithmetic builds a new function: $\displaystyle f(x)+g(x)$ , $\displaystyle f(x)-g(x)$ , $\displaystyle f(x)\cdot g(x)$ and $\displaystyle \frac{f(x)}{g(x)}$ .

The sign of a product and of a quotient comes from the signs: the same sign – positive, different signs – negative.
$\displaystyle \frac{f(x)}{g(x)}$ is not defined where $\displaystyle {g(x)}={0}$ .

In the example below, $\displaystyle {f(x)}={x-1}$ and $\displaystyle {g(x)}={x+2}$ :

$\displaystyle {f(x)\cdot g(x)}={(x-1)(x+2)}$ is negative where $\displaystyle -2<x<1$ , where $\displaystyle f$ and $\displaystyle g$ have different signs.
$\displaystyle {\frac{f(x)}{g(x)}}={\frac{x-1}{x+2}}$ is defined where $\displaystyle x\neq-2$ , and has the same signs as the product.

Composing functions

Composing functions puts one function inside the other: $\displaystyle f(g(x))$ . First work out $\displaystyle g(x)$ – the inner function – and put the result into $\displaystyle f$ – the outer function.

In the example below, $\displaystyle {f(x)}={x^2}$ and $\displaystyle {g(x)}={x-3}$ :

$\displaystyle {f(g(x))}={(x-3)^2}$ : first subtract 3, then square.
$\displaystyle {g(f(x))}={x^2-3}$ : first square, then subtract 3.
At $\displaystyle {x}={0}$ : $\displaystyle {f(g(0))}={9}$ and $\displaystyle {g(f(0))}={-3}$ . The order of composing changes the result.

The power of a function and the absolute value of a function are compositions too: $\displaystyle (x^2-5)^3$ is $\displaystyle x^3$ with $\displaystyle x^2-5$ put in it, and $\displaystyle |x^2-4|$ is $\displaystyle |x|$ with $\displaystyle x^2-4$ put in it.

Exercise 1

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

Mark the zeros of $\displaystyle h$ on the graph.

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

Where is $\displaystyle {h(x)}={-(x^2-2x-3)}$ ?
Explain why $\displaystyle (1,4)$ is a maximum point of $\displaystyle h$ .

Exercise 2

Solve the equation and the inequality.

Exercise 3

The functions are $\displaystyle {f(x)}={x^2+1}$ and $\displaystyle {g(x)}={2x-3}$ .

Write $\displaystyle f(g(x))$ without brackets.

$\displaystyle f(x)=x^{2}+1,\qquad g(x)=2x-3$

/ – 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

Write $\displaystyle g(f(x))$ without brackets.

$\displaystyle f(x)=x^{2}+1,\qquad g(x)=2x-3$

/ – 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

In each composition, which is the inner function and which the outer?
Is $\displaystyle {f(g(x))}={g(f(x))}$ ?

Exercise 4

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

Solve the inequality $\displaystyle f(x)\cdot g(x)<0$ .

What is the domain of $\displaystyle \frac{f(x)}{g(x)}$ ?
Solve the inequality $\displaystyle \frac{f(x)}{g(x)}\geq0$ .

Exercise 5

Each function is a composition of two functions.
Write the inner function and the outer function of each.

$\displaystyle {h(x)}={(x^2-5)^3}$
$\displaystyle {k(x)}={|x^3-8|}$
$\displaystyle {m(x)}={(2x+1)^2+3(2x+1)}$

Exercise 6

The function is $\displaystyle {f(x)}={|x^2-4|}$ .

Mark on the board the points where the graph of $\displaystyle f$ meets the line $\displaystyle {y}={3}$ .

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

\displaystyle g(x)=3

Solve the equation $\displaystyle {f(x)}={3}$ .

Solve the inequality $\displaystyle f(x)<3$ .
Check on the board: where is the graph below the line?

Formulas: calculus, grade 10 (5 units)

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Algebra

The square of a sum and of a difference
$\displaystyle (a\pm b)^2$ $\displaystyle =$ $\displaystyle a^2\pm2ab+b^2$
The difference of two squares
$\displaystyle a^2-b^2$ $\displaystyle =$ $\displaystyle (a-b)(a+b)$
The cube of a sum and of a difference
$\displaystyle (a\pm b)^3$ $\displaystyle =$ $\displaystyle a^3\pm3a^2b+3ab^2\pm b^3$
The sum and the difference of two cubes
$\displaystyle a^3\pm b^3$ $\displaystyle =$ $\displaystyle (a\pm b)(a^2\mp ab+b^2)$
The quadratic formula for $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
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 properties of a function

An even function – the graph is symmetric about the $\displaystyle y$ -axis
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle f(x)$
An odd function – the graph is symmetric about the origin
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle -f(x)$
The root of a square
$\displaystyle \sqrt{x^2}$ $\displaystyle =$ $\displaystyle |x|$
The absolute value
$\displaystyle |a|=\begin{cases}a & a\geq0\\-a & a<0\end{cases}$

Transformations of a graph

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

The derivative

The average rate of change from $\displaystyle x_1$ to $\displaystyle x_2$ – the slope of the secant
$\displaystyle \frac{f(x_2)-f(x_1)}{x_2-x_1}$
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 instantaneous speed – the derivative of the distance with respect to time
$\displaystyle v(t)$ $\displaystyle =$ $\displaystyle s'(t)$

Rules of differentiation

The derivative of a product of functions
$\displaystyle [f(x)\cdot g(x)]'$ $\displaystyle =$ $\displaystyle f'(x)\cdot g(x)+f(x)\cdot g'(x)$
The derivative of a quotient of functions
$\displaystyle \left[\frac{f(x)}{g(x)}\right]'$ $\displaystyle =$ $\displaystyle \frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{[g(x)]^2}$
The derivative of a composite function – the outer derivative times the inner derivative
$\displaystyle [f(u(x))]'$ $\displaystyle =$ $\displaystyle f'(u)\cdot u'(x)$
The derivative of $\displaystyle \frac{1}{x}$
$\displaystyle \left(\frac{1}{x}\right)'$ $\displaystyle =$ $\displaystyle -\frac{1}{x^2}$
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
$\displaystyle (\sqrt{f(x)})'$ $\displaystyle =$ $\displaystyle \frac{f'(x)}{2\sqrt{f(x)}}$ $\displaystyle f(x)>0$

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
$\displaystyle y-y_1=m(x-x_1)$

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
$\displaystyle f'(x)=0$

Asymptotes

A vertical asymptote – $\displaystyle a$ a zero of the denominator that is not a zero of the numerator
$\displaystyle x=a$
A horizontal asymptote – $\displaystyle b$ the value the function approaches as $\displaystyle x$ grows or falls without end
$\displaystyle y=b$