The rational function: domain and asymptotes

Calculus, grade 11 (4 units) Lesson 4 of 20Learning

The rational function: domain and asymptotes

A rational function

A rational function is one polynomial divided by another. In this course the numerator and the denominator are polynomials of degree at most 2:

$\displaystyle {f(x)}={\frac{2x+1}{x-3}}$ ;
$\displaystyle {f(x)}={\frac{x}{x^2-4}}$ ;
$\displaystyle {f(x)}={\frac{3x^2}{x^2-4}}$ .

The sum or difference of such a function and a polynomial, like $\displaystyle {f(x)}={x+\frac{4}{x}}$ , is investigated in the same ways.

The domain

There is no dividing by 0, so the domain is every $\displaystyle x$ where the denominator is not 0.
It is found like this:

solve the equation “the denominator $\displaystyle {}={0}$ ”;
its solutions are left out of the domain.

For example, in $\displaystyle {f(x)}={\frac{x}{x^2-4}}$ : $\displaystyle {x^2-4}={0}$ when $\displaystyle {x}={2}$ or $\displaystyle {x}={-2}$ , and the domain is $\displaystyle x\neq2$ , $\displaystyle x\neq-2$ .
The denominator $\displaystyle x^2+1$ has no zeros, so $\displaystyle \frac{x}{x^2+1}$ is defined for every $\displaystyle x$ .

A vertical asymptote

Near a zero $\displaystyle {x}={a}$ of the denominator, the denominator is close to 0 and the numerator is not. A number not close to 0, divided by a number close to 0, is very far from 0 – so the graph runs off upwards or downwards, and the line $\displaystyle {x}={a}$ is a vertical asymptote.
In this course the numerator is never 0 at the same point as the denominator, so every zero of the denominator gives a vertical asymptote.

A horizontal asymptote

When $\displaystyle x$ is very large (or very small), the value is set by the highest powers in the numerator and the denominator. For example, in $\displaystyle \frac{2x+1}{x-3}$ , when $\displaystyle {x}={1000}$ : $\displaystyle \frac{2001}{997}\approx2.007$ – almost $\displaystyle {\frac{2x}{x}}={2}$ .
So the horizontal asymptote depends on the degrees of the numerator and the denominator:

The numerator’s degree is lower than the denominator’s: the denominator grows faster, and the fraction comes close to 0 – the asymptote is $\displaystyle {y}={0}$ .
The same degree: the fraction comes close to the quotient of the coefficients of the highest power, $\displaystyle {y}={\frac{a}{b}}$ . In $\displaystyle \frac{3x^2}{x^2-4}$ : $\displaystyle {y}={\frac{3}{1}}={3}$ .
The numerator’s degree is higher, or a sum with a polynomial like $\displaystyle x+\frac{4}{x}$ : the values keep growing, and there is no horizontal asymptote.

Example

$\displaystyle {f(x)}={\frac{2x+1}{x-3}}$ .
In the example below we find the domain and the asymptotes:

The denominator is 0 when $\displaystyle {x}={3}$ , so the domain is $\displaystyle x\neq3$ .
The vertical asymptote: $\displaystyle {x}={3}$ .
The numerator and the denominator are of degree 1, and the coefficients of $\displaystyle x$ are 2 and 1, so the horizontal asymptote is $\displaystyle {y}={\frac{2}{1}}={2}$ .

\displaystyle f(x)=\frac{2x+1}{x-3}

\displaystyle x=3

\displaystyle y=2

The graph comes close to both dashed lines and never touches them: to the left of $\displaystyle {x}={3}$ it goes down, and to the right of it it goes up.

Exercise 1

$\displaystyle {f(x)}={\frac{x+1}{x^2-5x+6}}$ .

Solve the equation “the denominator $\displaystyle {}={0}$ ”, and write the domain of $\displaystyle f$ .

Write the equations of all the asymptotes of $\displaystyle f$ on the board.

\displaystyle f(x)=\frac{x+1}{x^{2}-5x+6}

Exercise 2

$\displaystyle {f(x)}={\frac{3x^2}{x^2-4}}$ .
Write the equations of the three asymptotes of $\displaystyle f$ on the board.

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

Exercise 3

$\displaystyle {f(x)}={x+\frac{4}{x}}$ .
Write the equation of the vertical asymptote of $\displaystyle f$ on the board.

\displaystyle f(x)=x+\frac{4}{x}

What is the domain of $\displaystyle f$ ? Work out $\displaystyle f(100)$ , and explain why $\displaystyle f$ has no horizontal asymptote.

\displaystyle f(100)

/ – 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

Exercise 4

Write the domain and the asymptotes of each function:

$\displaystyle {f(x)}={\frac{5}{x+4}}$
$\displaystyle {g(x)}={\frac{x^2+1}{x^2}}$
$\displaystyle {h(x)}={\frac{4x}{2x-1}}$
$\displaystyle {k(x)}={\frac{x-3}{x^2+9}}$

Exercise 5

$\displaystyle {f(x)}={\frac{2x-6}{x^2+1}}$ .
$\displaystyle f$ has no vertical asymptote: the denominator has no zeros.
Write the equation of its horizontal asymptote on the board.

\displaystyle f(x)=\frac{2x-6}{x^{2}+1}

Formulas: calculus, grade 11 (4 units)

Open as a page

Transformations of the graph of $f$

A horizontal shift: $\displaystyle p$ units to the right ( $\displaystyle p>0$ ) or to the left ( $\displaystyle p<0$ )
$\displaystyle y=f(x-p)$
A vertical shift: $\displaystyle k$ units up ( $\displaystyle k>0$ ) or down ( $\displaystyle k<0$ )
$\displaystyle y=f(x)+k$
A reflection in the $\displaystyle x$ -axis
$\displaystyle y=-f(x)$
A reflection in the $\displaystyle y$ -axis
$\displaystyle y=f(-x)$
A vertical stretch ( $\displaystyle a>1$ ) or squeeze ( $\displaystyle 0<a<1$ )
$\displaystyle y=a\cdot f(x)$
Absolute value: the part below the $\displaystyle x$ -axis is reflected above it
$\displaystyle y=|f(x)|$

The rational function

The horizontal asymptote when the numerator and the denominator have the same degree – $\displaystyle a$ and $\displaystyle b$ the coefficients of the highest power
$\displaystyle y=\frac{a}{b}$
The horizontal asymptote when the numerator’s degree is lower than the denominator’s
$\displaystyle y=0$
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
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle -f(x)$

Derivatives and the tangent

The derivative of a power
$\displaystyle \left(x^{n}\right)'=nx^{n-1}$
The derivative of a product
$\displaystyle \left(u\cdot v\right)'$ $\displaystyle =$ $\displaystyle u'\cdot v+u\cdot v'$
The derivative of a quotient
$\displaystyle \left(\frac{u}{v}\right)'$ $\displaystyle =$ $\displaystyle \frac{u'\cdot v-u\cdot v'}{v^{2}}$
The derivative of $\displaystyle \frac{c}{x}$
$\displaystyle \left(\frac{c}{x}\right)'$ $\displaystyle =$ $\displaystyle -\frac{c}{x^{2}}$
The derivative of a root – $\displaystyle u$ a function of $\displaystyle x$
$\displaystyle \left(\sqrt{u}\right)'$ $\displaystyle =$ $\displaystyle \frac{u'}{2\sqrt{u}}$
The equation of the tangent at the point $\displaystyle (x_1,y_1)$
$\displaystyle y-y_1$ $\displaystyle =$ $\displaystyle m(x-x_1)$
The slope of the tangent
$\displaystyle m$ $\displaystyle =$ $\displaystyle f'(x_1)$

Investigating a function with its derivative

The function rises
$\displaystyle f'(x)>0$
The function falls
$\displaystyle f'(x)<0$
A possible extremum point – check the sign of $\displaystyle f'$ on each side of it
$\displaystyle f'(x)=0$

Integrals and areas

The integral of a power
$\displaystyle \int x^{n}\,dx$ $\displaystyle =$ $\displaystyle \frac{x^{n+1}}{n+1}+C$ $\displaystyle n\neq-1$
The integral of $\displaystyle \frac{c}{(ax+b)^{2}}$
$\displaystyle \int\frac{c}{(ax+b)^{2}}\,dx$ $\displaystyle =$ $\displaystyle -\frac{c}{a(ax+b)}+C$
The definite integral – $\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int_{a}^{b}f(x)\,dx$ $\displaystyle =$ $\displaystyle F(b)-F(a)$
The area between two graphs, from $\displaystyle {x}={a}$ to $\displaystyle {x}={b}$
$\displaystyle S$ $\displaystyle =$ $\displaystyle \int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$