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Rational Functions and Their Domains - Prof. Frances Ellen M. Alvarado, Study notes of Algebra

University of Texas - Rio Grande Valley Algebra

Prof. Frances Ellen M. Alvarado

What a rational function is, provides examples to identify rational functions, and discusses how to find the domain of a rational function. The document also mentions vertical asymptotes.

Typology: Study notes

Pre 2010

Uploaded on 08/16/2009

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What is a Rational Function and What is it’s Domain?

Written by Frances E. M. Alvarado

March 19, 2006

So what’s a Rational Function? It is a Function written as

   

 

xQ

xP

xf 

where P(x) and Q(x) are Polynomial Functions. So which of the following are

Rational Functions?

Example 1: Which of the following are Rational Functions?

a.)

 

2x

4x

xf 





c.)

 

2x

x4

xf

3





b.)

 

2x

4x

xf 





d.)

 

2x

4

xf 



Did you guess which ones were? If not you may want to visit the Math 1334

section for their handouts on Rational Expressions and Equations.

Once we are able to recognize whether something is a Rational Function,

then we need to be able to find Domain, Range, Intercepts, f(any number),

and x when f(x) = any number, etc . In other words, everything we did with

all the previous functions.

We start with how to find the Domain (what value is or is not allowed for “x”).

As you may recall, we are not allowed to “divide by 0”. So that means our

denominator or bottom polynomial



0. So the best way to find out what

makes the denominator zero is to let the “Denominator = 0”, Solve, and the

tell the fractions “NANI NANI BOO BOO – you can’t have it”. This means the

Domain is “all real numbers except those that make the denominator zero”.

So if you were given

 

3x

9x

xf

2







, what would be the Domain of this

Function. Well the only number that will make the denominator zero would

be “x = 3”. So the Domain is “All real numbers except x = 3”. How did I find

that so fast – well simple I solved “x – 3 = 0”.

Can you find the Domain? Try the problems on the next page.

Example 2: Find the Domain of the following functions.

a.)

 

5

2x

xf 



c.)

 

5x

x3

xf 



b.)

 

6x5x

1x2

xf

2







d.)

 

7x5x

1x2

xf

2







Partial preview of the text

Download Rational Functions and Their Domains - Prof. Frances Ellen M. Alvarado and more Study notes Algebra in PDF only on Docsity!

What is a Rational Function and What is it’s Domain? Written by Frances E. M. Alvarado March 19, 2006

So what’s a Rational Function? It is a Function written as ^ ^

Q  x

P x f x  where P(x) and Q(x) are Polynomial Functions. So which of the following are Rational Functions? Example 1: Which of the following are Rational Functions?

a.)  

x 2 x 4 f x  

 c.) ^ ^

x 2 4 x f x 3  

b.) ^ ^

x 2 x 4 f x  

 d.) ^ ^

x 2 4 f x   Did you guess which ones were? If not you may want to visit the Math 1334 section for their handouts on Rational Expressions and Equations. Once we are able to recognize whether something is a Rational Function, then we need to be able to find Domain, Range, Intercepts, f(any number), and x when f(x) = any number, etc. In other words, everything we did with all the previous functions. We start with how to find the Domain (what value is or is not allowed for “x”). As you may recall, we are not allowed to “divide by 0”. So that means our

denominator or bottom polynomial  0. So the best way to find out what

makes the denominator zero is to let the “Denominator = 0”, Solve, and the tell the fractions “NANI NANI BOO BOO – you can’t have it”. This means the Domain is “all real numbers except those that make the denominator zero”.

So if you were given ^ ^

x 3 x 9 f x 2    (^) , what would be the Domain of this Function. Well the only number that will make the denominator zero would be “x = 3”. So the Domain is “All real numbers except x = 3”. How did I find that so fast – well simple I solved “x – 3 = 0”. Can you find the Domain? Try the problems on the next page. Example 2: Find the Domain of the following functions.

a.) ^ ^

5 x 2 f x 

 c.)  

x 5 3 x f x  

b.) ^ ^

x 5 x 6 2 x 1 f x 2   

 d.)  

x 5 x 7 2 x 1 f x 2    

Note: You have to know how to solve Linear, Quadratic, and Higher-Order Equations. Further notice, how this relates to what is normally referred to as a Restriction in Rational Equations. Once we find the Domain we can find something else called a “Vertical Asymptote”. What’s a Vertical Asymptote you ask? Very simple – it is a Vertical line that the graph approaches as you approach the Restrictions of the Domain.

So let’s look at a simple function first: ^ ^

x 3 4 x f x   In this function the Domain is “All real numbers except x = 3”. Interestingly enough “x = 3” is the Vertical Line the graph will approach from the left and the right. To convince yourself that this works – let x equal the following values and find the corresponding “y-values”. Then plot all the points you find and play connect the dots. Notice how close to “x = 3” you get. Here are the x-values I would suggest you use: Let x = 0, 1, 2, 2.8, 2.9, 3, 3.01, 3.1, 4, 5, 6. Is this enough to graph – No but it’s a beginning. Okay the graph should look something like this: Y axis Vertical Asymptote x axis

Rational Functions and Their Domains - Prof. Frances Ellen M. Alvarado, Study notes of Algebra

Related documents

Partial preview of the text

Download Rational Functions and Their Domains - Prof. Frances Ellen M. Alvarado and more Study notes Algebra in PDF only on Docsity!

So what’s a Rational Function? It is a Function written as ^ ^

Q  x

a.)  

 c.) ^ ^

b.) ^ ^

 d.) ^ ^

denominator or bottom polynomial  0. So the best way to find out what

So if you were given ^ ^

a.) ^ ^

 c.)  

b.) ^ ^

 d.)  

So let’s look at a simple function first: ^ ^