Rational functions are functions that have a fraction with a polynomial in the denominator and a polynomial in the numerator. In order to graph these functions, it is necessary to determine their asymptotes.

Here we look at a summary of rational functions. Let's look at how to graph rational functions and how to find asymptotes. In addition, we will look at several examples of rational functions with answers to fully understand the process used to find asymptotes and graph these types of functions.

##### ALGEBRA

**relevant for**…

Learn about rational functions using examples.

**see examples**

**Contents**

- Summary of rational functions
- Examples with solutions of rational function problems
- also see

##### ALGEBRA

**relevant for**…

Learn about rational functions using examples.

**see examples**

## Summary of rational functions

A rational function is a function that can be written as a fraction of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function need necessarily be rational numbers.

A function of the variable*X*is considered a rational function only if it can be written in the form:

$latex f(x)=\frac{P(x)}{Q(x)}$

Wo,*P*e*Q*are polynomial functions and $latex Q(x)$ is not zero.

### Graphs of Rational Functions

To graph rational functions, we follow these steps:

**Step 1:**Find the intercepts, if any. O*j*-intercept is the point $latex(0, ~f(0))$ and we find it*X*- intercepts by setting the numerator to an equation equal to zero and solving*X*.

** Step 2:**We find the vertical asymptotes by setting the denominator equal to zero and solving.

** Level 3:**If it exists, we find the horizontal asymptote using the details below about asymptotes.

** Step 4:**Vertical asymptotes divide the graph into several areas. We need to find multiple points in each of the regions to determine the overall shape the graph will take.

** Step 5:**We draw the graph going through all the found points.

### find asymptotes

If we have the rational function $latex \frac{a{{x}^n}+\cdots}{b{{x}^m}+\cdots}$ , where*N*means the largest exponent of the numerator and*M*means the largest denominator of the exponent. We can find asymptotes as follows:

1. If the denominator at $latex x=a$ is zero and the denominator at $latex x=a$ is non-zero, the graph has a vertical asymptote at $latex x=a$.

2. Se $latex n<m$, o*X*-Axis is the horizontal asymptote.

3. If $latex n=m$ then the line $latex y=\frac{a}{b}$ is the horizontal asymptote.

4. If $latex n>m$ then there are no horizontal asymptotes

## Examples with solutions of rational function problems

The following rational function problems are solved using the process described above. Try to solve the tasks yourself before you see the answer.

**EXAMPLE 1**

Draw the graph of the rational function $latex f(x)= \frac{-3}{x-1}$.

##### Solution

**Step 1:**We start by finding the intersections of the function:

- Ö
*j*section is the point $latex(0,~ f(0))=(0, 3)$. - For
*X*-Interceptions, we set the counter to zero and solve. Here, however, the numerator is the constant -3, so it has no zeros. Therefore the function does not have*X*-intercepts.

**Step 2:**To find the vertical asymptotes, we set the denominator to zero and solve:

$latex x-1=0$

$latex x=1$

We have a vertical asymptote at $latex x=1$.

**Level 3:**The largest exponent of*X*in the denominator is 1, which is greater than the exponent of*X*in the counter (0). So the*X*-Axis is the horizontal asymptote.

**Step 4:**We only have one vertical asymptote, so we have two areas in the chart: $latex x>1$ and $latex x<1$.

We need a point in each region to determine whether it is above or below the horizontal asymptote. Therefore we can use:

$latex f(0)=3$ ⇒ $latex (0, ~3)$

$latex f(2)=-3$ ⇒ $latex (2, ~-3)$

**Step 5:**Here is the graph of the function:

**EXAMPLE 2**

Draw the rational function $latex f(x)=\frac{4-2x}{1-x}$.

##### Solution

**Step 1:**We need to find the sections of the function:

- Ö
*j*-intercept is the point $latex(0, ~f(0))=(0, ~4)$. - we found them
*X*-Sections by setting the counter to zero and solving:

$latex 4-2x=0$

$latex -2x=-4$

$latex x= 2$

Ö*X*-Intercept ist $latex x=2$.

**Step 2:**We find the vertical asymptotes by setting the denominator to zero and solving:

$latex 1-x=0$

$latex x=1$

We have a vertical asymptote at $latex x=1$.

**Level 3:**The greatest representatives of*X*Both the denominator and the numerator are the same. So the horizontal asymptote is equal to the coefficient of*X*in the numerator divided by the coefficient of*X*no denominator:

$latex y=\frac{-2}{-1}=2$

**Step 4:**We have a vertical asymptote, so we only have two areas in the chart: $latex x>1$ and $latex x<1$.

We need to find a point in each region to know if it's above or below the horizontal asymptote. So we use:

$latex f(0)=4$ ⇒ $latex (0, ~4)$

$latex f(2)=0$ ⇒ $latex (2, ~0)$

**Step 5:**With the obtained points we plot the function:

**EXAMPLE 3**

Draw the rational function $latex f(x)=\frac{4}{{x}^2}+x-2}$.

##### Solution

**Step 1:**We need to find the sections of the function:

- Ö
*j*-intercept is the point $latex(0, ~f(0))=(0, ~-2)$. - we found them
*X*-intercepts by setting the counter equal to zero and resolving. In this case, the numerator is the constant 4, so we don't have zeros and the function doesn't*X*-intercepts.

**Step 2:**We set the denominator to zero and solve to find the vertical asymptotes:

$latex {{x}^2}+x-2=0$

$latex (x+2)(x-1)=0$

We have the vertical asymptotes $latex x=1$ and $latex x=-2$.

**Level 3:**In this function is the largest exponent of*X*in the denominator is greater than the exponent of*X*in the counter (0). So the*X*-Axis is the horizontal asymptote.

**Step 4:**We have two vertical asymptotes, i.e. three areas in the diagram: $latex x<-2$, $latex -2<x<1$ and $latex x>1$.

We need a point in each region to determine whether it is above or below the horizontal asymptote. The middle section is a bit trickier, so we need some points near the vertical asymptotes. Therefore we can use:

$latex f(-3)=1$ ⇒ $latex (-3, ~1)$

$latex f(-1)=-2$ ⇒ $latex (-1, ~-2)$

$latex f(0)=-2$ ⇒ $latex (0, ~-2)$

$latex f(2)=1$ ⇒ $latex (2, ~1)$

**Step 5:**Here is the graph of the function:

**EXAMPLE 4**

Find the graph of the rational function $latex f(x)=\frac{{{x}^2}-4}{{{x}^2}-4x}$.

##### Solution

**Step 1:**The intersection points of the rational function are:

- If we use $latex x=0$ we get division by zero, so the function has no a
*j*-intercept. However, we have already found a vertical asymptote. - Ö
*X*-Interruptions are:

$latex {{x}^2}-4=0$

$latex x=\pm 2$

So we have two*X*-intercepts.

**Step 2:**We've already found one vertical asymptote, but there could be more. Therefore we set the denominator to zero and solve:

$latex {{x}^2}-4x=x(x-4)=0$

$latex x=0$ und $latex x=4$

We got two vertical asymptotes in $latex x=0$ and $latex x=4$.

**Level 3:**The largest exponent of*X*in the numerator is 2, similar to the numerator. So we have a horizontal asymptote at:

$latex y=\frac{1}{1}=1$

**Step 4:**We have the regions: $latex x<0$, $latex 0<x<4$ and $latex x>4$.

One of*X*-intercepts is in the left pane, so we don't need a dot there. The other*X*-intercept is in the middle range, but we need more points to determine its behavior. Also, we need a point in the right region. So we have:

$latex f(1)=1$ ⇒ $latex (1,~1)$

$latex f(3)=-\frac{5}{3}$ ⇒ $latex (3, ~-\frac{5}{3})$

$latex f(5)=\frac{21}{5}$ ⇒ $latex (5, ~\frac{21}{5})$

**Step 5:**Here is the graph of the function:

**EXAMPLE 5**

Draw the function $$f(x)= \frac{4{x}^2}-36}{{x}^2}-2x-8}$$

##### Solution

**Step 1:**We start with the interceptions of the function:

- Ö
*j*section is the point $latex(0, f(0))=(0, \frac{9}{2})$. - Ö
*X*-Interruptions are:

$latex 4{{x}^2}-36=0$

$latex 4{{x}^2}=36$

$latex {{x}^2}=9$

$latex x= \pm3$

Ö*X-*Sections are (-3, 0) and (3, 0).

**Step 2:**To find the vertical asymptotes, we set the denominator to zero and solve:

$latex {{x}^2}-2x-8=0$

$latex (x+2)(x-4)=0$

We have two vertical asymptotes in $latex x=-2$ and $latex x=4$.

**Level 3:**The largest exponent of*X*in both the denominator and the numerator is 2. So the horizontal asymptote is:

$latex y= \frac{4}{1}=4$

**Step 4:**We have two vertical asymptotes, i.e. the regions: $latex x<-2$, $latex -2<x<4$ and $latex x>4$.

We need a point in each region, but we need some points in the middle region. Therefore we can use:

$latex f(-3)=0$⇒ $latex (-3, 0)$

$latex f(-1)= \frac{32}{5}$⇒ $latex (-1, \frac{32}{5})$

$latex f(3)=0$⇒ $latex (3, 0)$

$latex f(5)=\frac{64}{7}$⇒ $latex (5, \frac{64}{7})$

**Step 5:**Here is the graph of the function:

## also see

Want to learn more about rational functions? Check out these pages:

- Applications of rational functions
- Find asymptotes of a function

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## FAQs

### What is an example of rational function problem? ›

**One person can complete a task 8 hours sooner than another person**. Working together, both people can perform the task in 3 hours. How many hours does it take each person to complete the task working alone?

**What is an example of rational function with answer? ›**

For example, **f(x) = (x ^{2} + x - 2) / (2x^{2} - 2x - 3)** is a rational function and here, 2x

^{2}- 2x - 3 ≠ 0. We know that every constant is a polynomial and hence the numerators of a rational function can be constants also. For example, f(x) = 1/(3x+1) can be a rational function.

**What is the 5 example of rational function? ›**

Answer: Recall that a rational function is defined as the ratio of two real polynomials with the condition that the polynomial in the denominator is not a zero polynomial. f(x)=P(x)Q(x) f ( x ) = P ( x ) Q ( x ) , where Q(x)≠0. An example of a rational function is: **f(x)=x+12x2−x−1z**.

**What solves problems involving rational functions? ›**

To solve an equation involving rational functions, we **cross multiply the numerators and denominators.** **Then we move all our terms to one side.** **Then we use our algebra skills to solve**. To solve an inequality involving rational functions, we set our numerator and denominator to 0 and solve them separately.

**What are real life examples of rational functions? ›**

**Work problems**

A “work problem” is an example of a real life situation that can be modeled and solved using a rational equation. Work problems often ask you to calculate how long it will take different people working at different speeds to finish a task.

**What is an example of using rational equations to solve real world problems? ›**

A “**work problem**” is an example of a real life situation that can be modeled and solved using a rational equation. Work problems often ask you to calculate how long it will take different people working at different speeds to finish a task.

**What are 10 examples of rational? ›**

number which is represented in the form of p/q where p and q are coprime integers and q is not equal to 0. example : **2/10,2/7,root4,5/1,0/10,6/7,2/3,5/7,1/2,5/4**.

**How do you write a rational function from a word problem? ›**

**How to Solve a Word Problem with Rational Equations**

- Read the problem carefully and determine what you are asked to find.
- Assign a variable to represent the unknown.
- Write out an equation that describes the given situation.
- Solve the equation.
- State the answer using a nice clear sentence.

**Which is an example of a rational equation? ›**

Equations that contain rational expressions are called rational equations. For example, **2x+14=7x 2 x + 1 4 = 7 x** is a rational equation. Rational equations can be useful for representing real-life situations and for finding answers to real problems.

**What are the 5 examples of rational inequality? ›**

A rational inequality is an inequality that contains a rational expression. Inequalities such as **3 2 x > 1 , 2 x x − 3 < 4 , 2 x − 3 x − 6 ≥ x , 3 2 x > 1 , 2 x x − 3 < 4 , 2 x − 3 x − 6 ≥ x , and 1 4 − 2 x 2 ≤ 3 x 1 4 − 2 x 2 ≤ 3 x** are rational inequalities as they each contain a rational expression.

### What is a rational problem? ›

**problem solving based on reasoning that is generally agreed to be correct, optimal, or logical**.