Wize University Calculus 1 Textbook > Pre-Calculus (Review)

Rational Functions

Rational Functions

A rational function can be expressed as:

\boxed{f(x)=\displaystyle\frac{ax^n+bx^{n-1}+...}{cx^m+dx^{m-1}+...}}

where $n$ is the largest exponent in the numerator and $m$ is the largest exponent in the denomiator.

Properties of Rational Functions

X-Intercepts can be found by setting $f(x)=0$ and solving for $x$
Vertical asymptotes can be found by setting the denominator to 0 and solving for $x$
Horizontal asymptotes:
If $\underline{n<m}$ : There is a horizontal asymptote at $y=0$
If $\underline{n=m}$ : There is a horizontal asymptote at $y=\displaystyle\frac{a}{c}$
If $\underline{n>m}$ : There is no horizontal asymptote. There may be an oblique/diagonal asymptote.
If factoring the numerator & denominator of $f(x)$ , any terms that cancel out indicates where a missing point/hole in $f(x)$
The domain is the set of all real numbers except where there are vertical asymptotes & missing points/holes
The range is the set of all real numbers except where there are horizontal asymptotes in most cases

Example 1

Let's look at $f(x)=\displaystyle\frac{2x-3}{x+1}$ .

$x$ -Intercept: $\Bigg(\displaystyle\frac{3}{2},0\Bigg)$
Vertical Asymptote: $x=-1$
Horizontal Asymptote: $y=2$
Missing Points/Holes: None
Intervals of Increasing: $(-\infin,-1)~\cup~(-1,\infin)$
Intervals of Decreasing: NA
Domain: $(-\infin,-1)~\cup~(-1,\infin)$
Range: $(-\infin,2)~\cup~(2,\infin)$

Solving Rational Equations

Rational equations are a type of equation containing at least one rational term, $\boxed{\displaystyle\frac{f(x)}{g(x)}}$ , where $f(x)~\&~g(x)$ are continuous polynomials.

Rational equations can contain non-permissible values (NPV) that identify the restrictions of the domain. They identify the value of the variable that gives a $0~$ in the denominator.

Example

Solve $\displaystyle\frac{2}{x-1}=\displaystyle\frac{5}{x+2}$ , stating any NPV's.

Step 1.

Identify any non-permissible values.

The NPV's are the vertical asymptotes.

Therefore, $x\neq{-2,1}$ otherwise there is a total of $0$ in the denominator.

Step 2.

Find the lowest common denominator and multiply each term by it.

The lowest common denominator is $(x-1)(x+2)$ .

$\Bigg(\displaystyle\frac{2}{x-1}=\displaystyle\frac{5}{x+2}\Bigg)\times(x-1)(x+2)~~~{\color{red}\rightarrow}~~~~2(x+2)=5(x-1)$

Step 3.

Solve for $\green{'x'.}$

$\begin{array}{} 2(x+2)&=&5(x-1)\\\\ 2x+4&=&5x-5\\\\ 3x&=&9\\\\ \therefore~x&=&3 \end{array}$

Step 4.

Verify.

$\begin{array}{} \displaystyle\frac{2}{x-1}&=&\displaystyle\frac{5}{x+2}\\\\ \displaystyle\frac{2}{3-1}&=&\displaystyle\frac{5}{3+2}\\\\ \displaystyle\frac{2}{2}&=&\displaystyle\frac{5}{5}\\\\ 1&=&1&\checkmark \end{array}$

Example: Solving Rational Equations

Solve $\displaystyle\frac{3x}{x+5}+\frac{1}{x-2}=\displaystyle\frac{7}{x^2+3x-10}$ , stating any NPV's.

Step 1.

Identify any non-permissible values.

The NPV's are the vertical asymptotes.

$\begin{array}{rccccccc} (x+5)&=&0&&\rightarrow&&\therefore~x&=&-5\\\\ (x-2)&=&0&&\rightarrow&&\therefore~x&=&2\\\\ x^2+3x-10=(x+5)(x-2)&=&0&&\rightarrow&&\therefore~x&=&-5, 2 \end{array}$

Therefore, $x\neq{-5,2}$ otherwise there is a total of $0$ in the denominator.

Step 2.

Find the lowest common denominator and multiply each term by it.

The lowest common denominator is $(x+5)(x-2)$ .

$\Bigg(\displaystyle\frac{3x}{x+5}+\frac{1}{x-2}=\displaystyle\frac{7}{x^2+3x-10}\Bigg)\times(x+5)(x-2)~~~~~~{\color{red}\rightarrow}~~~~~~3x(x-2)+(x+5)=7$

Step 3.

Solve for $\green{'x'.}$

$\begin{array}{rcl} 3x(x-2)+(x+5)&=&7\\\\ 3x^2-6x+x+5&=&7\\\\ 3x^2-5x-2&=&0\\\\ (3x^2-6x)+(x-2)&=&0\\\\ (3x+1)(x-2)&=&0 \\\\\therefore~x&=&-\displaystyle\frac{1}{3}, 2 \end{array}$

According to the NPV's, $x\neq2$ .

Therefore, $x=2$ is an extraneous solution.

Step 4.

Verify.

$\begin{array}{rcl} \displaystyle\frac{3x}{x+5}+\frac{1}{x-2}&=&\displaystyle\frac{7}{x^2+3x-10}\\\\ \displaystyle\frac{3(-\frac{1}{3})}{(-\frac{1}{3})+5}+\frac{1}{(-\frac{1}{3})-2}&=&\displaystyle\frac{7}{(-\frac{1}{3})^2+3(-\frac{1}{3})-10} \\\\-\displaystyle\frac{3}{14}-\displaystyle\frac{3}{7}&=&-\displaystyle\frac{9}{14}\\\\ -\displaystyle\frac{9}{14}&=&-\displaystyle\frac{9}{14}~~~~~~~~~\checkmark \end{array}$

Therefore, $x=-\displaystyle\frac{1}{3}$ is the only solution.

Example: Rational Functions

A graph of the function $f(x)=\displaystyle{\frac{x^2-6x+8}{x^2-3x-4}}$ is sketched below:

Identify the equations for any vertical & horizontal asymptotes, and missing points/holes.

Factor $f(x)$ :

$\begin{array}{rcl} f(x)&=&\displaystyle{\frac{x^2-6x+8}{x^2-3x-4}}\\\\ f(x)&=&\displaystyle{\frac{(x-4)(x-2)}{(x-4)(x+1)}}\\\\ f(x)&=&\displaystyle{\frac{x-2}{x+1}} \end{array}$

Vertical Asymptotes:

Let $x+1=0$ .

The vertical asymptote is at $x=-1.$

Horizontal Asymptote:

The degree of the numerator is equivalent to the degree in the denominator.

Therefore, $y=1$ .

Missing Points/Holes

Missing points/holes exist whenever terms in the numerator and denominator cancel.

Since the term $(x-4)$ was eliminated, there is a missing point/hole at $x=4.$

Given the equation $\displaystyle\frac{2}{x-3}-\frac{3}{4-x}=\displaystyle\frac{2x-2}{x^2-x-12}$ , answer the following questions.

Solve for $x$ .

$x=$ (state the larger of the values)$x=$ (state the smaller of the values)

I don't know

Identify the vertical asymptotes, the horizontal asymptotes, the x-intercepts, and any missing points of:

$f(x)=\displaystyle\frac{x^2-11x+18}{x^2-3x+2}$

A) Vertical Asymptote $x=$B) Horizontal/Oblique Asymptote $y=$C) $x$-Intercepts $x=$D) Missing Points (if there are none, put none) $x=$

I don't know

Extra Practice

Rational Functions

When $x = 1000$ the function

f(x) = \frac{3x^9 + 7x^3 - 10}{5x^8 -3x^7 + 2x}

is closest to which number?

Rational Functions

When $x = 0.01$ the function

f(x) = \frac{3x^9 + 7x^3 - 10}{5x^8 -3x^7 + 2x}

is closest to which number?

Evaluating Rational Functions at a Point

When $x = -100$ , the function

f(x) = \frac{-3 x^{16} - 2 x^{10} - x^3 + 2}{15 x^{13} + 2 x^{10} - x^5 + 3}

is closest to which number?

Factoring and Cancelling terms in a Rational Function

When $x = 2.1$ , the function

f(x) = \frac{3x^2 - 3x - 6}{x^3 - 2x^2 - x + 2}

is closest to which number?

Rational Functions

Solve the following inequality for $x$

\frac{x-1}{x+1} \leq \frac{x+2}{x-1}

Rational Functions

Solve the following inequality: $\frac{3x+1}{x-2}\leq \frac{2x-1}{x+2}$

$\bf{Q:}$ Find $f(g(x))$ and its domain if $f\left(x\right)=\dfrac{1}{x-1}$ and $g\left(x\right)=\dfrac{1}{x+2}$

Solve the following inequality: $\dfrac{2x-1}{x-5}\ge\dfrac{x+1}{x+5}$

Rational Functions

Find all $x$ that satisfy $\dfrac{x}{x-3}\ge4$

Rational Functions: Inequalities with Fractions

Find all $x$ that satisfy $\dfrac{x+3}{x^2+8x+15}\le0$

Rational Functions: Slant Asymptote

Find the slant asymptote to the curve $\displaystyle y=\frac{x^3+x^2+1}{x^2+1}$ .

Rational Functions

$\bf{Q:}$ Find the exact list of vertical asymptotes of $\displaystyle f\left(x\right)=\frac{x-1}{x^2+x}$

Rules of Exponents

Simplify: $-\dfrac{5x^2y^a}{3^3z^5}\cdot\dfrac{3^7x^by}{15z^{-2}}$

Rational Functions

Match the following five functions with the above graphs.

Rational Functions

Which of the following graphs is a good approximation of $f(x) = \frac{x^2 - 3x}{x + 6}$ for $x$ on the small interval $[-1/10, 1/10]$

Rational Functions

Practice: Rational Functions

Match the rational function with the correct sketch of its graph.

Rational Functions

Practice: Rational Functions

Match the rational function with the correct sketch of its graph.

Rational Functions

Practice: Rational Functions

Identify the vertical asymptotes, the horizontal asymptotes, the x-intercepts, and any missing points of:

$f(x)=\dfrac{2x^2+x-1}{2x^2+5x-3}$

Rational Functions

Practice: Rational Functions

Identify the vertical asymptotes, the horizontal asymptotes, the x-intercepts, and any missing points of:

$f(x)=\dfrac{3x^2+14x+8}{x^2-2x-14}$

Rational Functions

Practice: Rational Functions

Match the equation for the rational function with its appropriate properties.

Rational Functions

Practice: Rational Functions

Match the equation for the rational function with its appropriate properties.

Graphing Rational Functions

Practice: Graphing Rational Functions

A rational function has the following properties:

A horizontal asymptote at $y=0$ .

Graphing Rational Functions

Practice: Graphing Rational Functions

A rational function has the following properties:

A horizontal asymptote at $y=0$ .

Graphing Rational Functions

Practice: Graphing Rational Functions

Sketch a graph of $f(x)=\dfrac{-x+9}{x+3}$ , finding all asymptotes, intercepts, missing points, and positive & negative intervals.

Graphing Rational Functions

Practice: Graphing Rational Functions

Sketch a graph of $f(x)=\dfrac{x^2-25}{x+5}$ , finding all asymptotes, x-intercepts, missing points, and positive & negative intervals.

Graphing Rational Functions

Practice: Graphing Rational Functions

Find a rational function of the form $f(x)=\displaystyle\frac{ax+b}{cx+d}$ that describes the following:

Vertical asymptote at $x=-2$

Graphing Rational Functions

Practice: Graphing Rational Functions

Find a rational function of the form $f(x)=\displaystyle\frac{ax+b}{cx+d}$ that describes the following:

Vertical asymptote at $x=-1$

Graphing Rational Functions

Practice: Graphing Rational Functions

Write a rational function with the following properties:

An oblique asymptote at $y=2x-7$

Graphing Rational Functions

Practice: Graphing Rational Functions

Write a rational function with the following properties:

An oblique asymptote at $y=-3x+1$

Solving Rational Equations

Solve $\dfrac{2x-3}{x^2-3}=\dfrac{5}{x+1}$ .

Solving Rational Equations

Solve for $x$ :

$\dfrac{5}{2x-1}-\dfrac{7}{3x+2}=\dfrac{9}{6x^2+x-2}$

Solving Rational Equations

Practice: Solving Rational Equations

True or false:

$\dfrac{17}{7}$ is a solution to $-5=\dfrac{2x+3}{x-4}$ .

Solving Rational Equations

Practice: Solving Rational Equations

True or false:

$\dfrac{81}{26}$ is a solution to $10=\dfrac{4x-1}{3x+8}$ .

Solving Rational Equations

Practice: Solving Rational Equations

Given the equation $\dfrac{4}{x+1}-\dfrac{1}{x+3}=\dfrac{x+2}{x^2+4x+3}$ , answer the following questions.

Solving Rational Equations

Practice: Solving Rational Equations

Given the equation $\dfrac{4x}{x-5}+\dfrac{3}{x+2}=\dfrac{x}{x^2-3x-10}$ , answer the following questions.

Solving Rational Equations

Practice: Solving Rational Equations

Let $f(x)=\dfrac{x^2-4}{x-2}$ .

If $f(x)=x^2-5x-6$ , then what is $x?$

Solving Rational Equations

Practice: Solving Rational Equations

Let $f(x)=\dfrac{4x^2-25}{2x+5}$ .

If $f(x)=4x^2-2x+1$ , then what is $x?$

Rational Functions

Let $f(x)=\dfrac{x^2+5x+6}{x-4}$ be a rational function.

Rules of Exponents

Simplify: $-\dfrac{5x^2y^a}{3^3z^5}\cdot\dfrac{3^7x^by}{15z^{-2}}$

Rational Functions

Find all $x$ that satisfy $\dfrac{x}{x-3}\ge4$

Rational Functions: Inequalities with Fractions

Find all $x$ that satisfy $\dfrac{x+3}{x^2+8x+15}\le0$

Solve the following inequality: $\dfrac{2x-1}{x-5}\ge\dfrac{x+1}{x+5}$

$\bf{Q:}$ Find $f(g(x))$ and its domain if $f\left(x\right)=\dfrac{1}{x-1}$ and $g\left(x\right)=\dfrac{1}{x+2}$

Rational Functions: Slant Asymptote

Find the slant asymptote to the curve $\displaystyle y=\frac{x^3+x^2+1}{x^2+1}$ .

Rational Functions

$\bf{Q:}$ Find the exact list of vertical asymptotes of $\displaystyle f\left(x\right)=\frac{x-1}{x^2+x}$

Rational Functions: Slant Asymptote

Find the slant asymptote to the curve $\displaystyle y=\frac{x^3+x^2+1}{x^2+1}$ .

Rational Functions

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