Find the domain of the rational function. Then, write it in lowest terms. f ( x ) = x 2 + 9 x − 3 f\left(x\right)=\frac{x^2+9}{x-3} f ( x ) = x − 3 x 2 + 9

{x∣x≠3}, f ( x ) = x 2 + 9 x − 3 f\left(x\right)=\frac{x^2+9}{x-3} f ( x ) = x − 3 x 2 + 9

{x∣x≠0}, f ( x ) = 1 x − 3 f\left(x\right)=\frac{1}{x-3} f ( x ) = x − 3 1

{x∣x≠−3}, f ( x ) = x 2 + 9 x − 3 f\left(x\right)=\frac{x^2+9}{x-3} f ( x ) = x − 3 x 2 + 9

{x∣x≠3}, f ( x ) = x + 3 f\left(x\right)=x+3 f ( x ) = x + 3

Find the domain of the rational function. Then, write it in lowest terms. f ( x ) = 6 x 5 2 x 2 − 8 f\left(x\right)=\frac{6x^5}{2x^2-8} f ( x ) = 2 x 2 − 8 6 x 5

{ x ∣ x ≠ 2 , − 2 } , f ( x ) = 3 x 5 x 2 − 4 {\left\lbrace x|x\ne2,-2\right\rbrace},f(x)=\frac{3x^5}{x^2-4} { x ∣ x  = 2 , − 2 } , f ( x ) = x 2 − 4 3 x 5

{ x ∣ x ≠ 2 } , f ( x ) = 3 x 5 x 2 − 4 {\left\lbrace x|x\ne2\right\rbrace},f(x)=\frac{3x^5}{x^2-4} { x ∣ x  = 2 } , f ( x ) = x 2 − 4 3 x 5

{ x ∣ x ≠ 2 } , f ( x ) = 3 x 5 x 2 − 8 {\left\lbrace x|x\ne2\right\rbrace},f(x)=\frac{3x^5}{x^2-8} { x ∣ x  = 2 } , f ( x ) = x 2 − 8 3 x 5

What is a rational function in algebra?

A rational function in algebra is a function that can be expressed as the ratio of two polynomials. It is written in the form p q , where p and q are polynomials. The domain of a rational function is all real numbers except where the denominator q is zero, as division by zero is undefined.

How do you simplify a rational function?

To simplify a rational function, factor both the numerator and the denominator to identify and cancel out common factors. For example, consider the function x +5 x 2 -25 . The numerator is already factored, and the denominator is a difference of squares, which factors to x +5 x -5 . Cancel the common factor x +5 to get the simplified function 1 x -5 .

5. Rational Functions

Introduction to Rational Functions

5. Rational Functions

Introduction to Rational Functions: Videos & Practice Problems

Video Lessons Practice Worksheet

Topic summary

Rational functions are formed by placing polynomials in the numerator and denominator, represented as p(x) / q(x). The domain is restricted where the denominator equals zero, requiring careful evaluation. For example, for the function f(x) = x + 5x² - 25, the domain excludes 5 and -5. Simplifying involves factoring and canceling common terms, ensuring domain restrictions are identified before simplification.

concept

Intro to Rational Functions

Video duration:

Intro to Rational Functions Video Summary

Rational functions are formed by taking a polynomial in the numerator and another polynomial in the denominator, expressed as p(x) / q(x). Understanding rational functions builds on prior knowledge of polynomial functions and rational equations. A key aspect of working with these functions is recognizing that the denominator cannot equal zero, as this would make the function undefined. This restriction leads to the concept of domain restrictions, which define the set of allowable values for x.

To determine the domain of a rational function, one must set the denominator equal to zero and solve for x. For example, for the function f(x) = 3 / (3x - 12), setting the denominator to zero gives:

3x - 12 = 0

Solving this yields x = -4, indicating that the domain is all real numbers except x = -4.

In another example, for f(x) = (x + 5) / (x² - 25), setting the denominator to zero results in:

x² - 25 = 0

Factoring gives (x - 5)(x + 5) = 0, leading to restrictions of x = 5 and x = -5. Thus, the domain is all real numbers except x = 5 and x = -5.

Another important skill when working with rational functions is simplifying them to their lowest terms. This involves factoring both the numerator and the denominator and canceling any common factors. For instance, in the function 3 / (3x + 12), factoring the denominator gives:

3 / (3(x + 4))

Canceling the common factor of 3 results in the simplified function 1 / (x + 4).

In the case of (x + 5) / (x² - 25), factoring the denominator as (x + 5)(x - 5) allows for cancellation of the common factor (x + 5), simplifying to 1 / (x - 5). However, it is crucial to find the domain before simplifying, as simplification may obscure additional restrictions. In this example, the domain restriction of x = -5 would be lost if simplification occurred first.

In summary, mastering rational functions involves understanding domain restrictions, simplifying expressions, and recognizing the importance of maintaining all restrictions throughout the process. This foundational knowledge is essential for further exploration of rational functions and their applications.

Study Smarter with Worksheets.

Follow along with each video using our printable worksheets

Problem

Find the domain of the rational function. Then, write it in lowest terms.
$f\left(x\right)=\frac{x^2+9}{x-3}$

{x∣x≠0}, $f\left(x\right)=\frac{1}{x-3}$

{x∣x≠3}, $f\left(x\right)=\frac{x^2+9}{x-3}$

{x∣x≠−3}, $f\left(x\right)=\frac{x^2+9}{x-3}$

{x∣x≠3}, $f\left(x\right)=x+3$

Problem

Find the domain of the rational function. Then, write it in lowest terms.
$f\left(x\right)=\frac{6x^5}{2x^2-8}$

${\left\lbrace x|x\ne2,-2\right\rbrace},f(x)=\frac{3x^5}{x^2-4}$

${\left\lbrace x|x\ne2,-2\right\rbrace},f(x)=\frac{6x^5}{2x^2-8}$

${\left\lbrace x|x\ne2\right\rbrace},f(x)=\frac{3x^5}{x^2-4}$

${\left\lbrace x|x\ne2\right\rbrace},f(x)=\frac{3x^5}{x^2-8}$

Do you want more practice?

More sets

Introduction to Rational Functions

5. Rational Functions

4 problems

Topic

Callie

5. Rational Functions

3 topics 7 problems

Chapter

Callie

Go over this topic definitions with flashcards

More sets

Introduction to Rational Functions definitions
5. Rational Functions
15 Terms

Here’s what students ask on this topic:

A rational function in algebra is a function that can be expressed as the ratio of two polynomials. It is written in the form $\frac{p}{q}$ , where $p$ and $q$ are polynomials. The domain of a rational function is all real numbers except where the denominator $q$ is zero, as division by zero is undefined.

To find the domain of a rational function, identify the values of $x$ that make the denominator zero. Set the denominator equal to zero and solve for $x$ . These values are excluded from the domain. For example, for the function $\frac{x +5}{x^{2} -25}$ , set $x^{2} -25=0$ and solve to find $x =5 and x =-5$ . The domain is all real numbers except $x =5 and x =-5$ .

To simplify a rational function, factor both the numerator and the denominator to identify and cancel out common factors. For example, consider the function $\frac{x +5}{x^{2} -25}$ . The numerator is already factored, and the denominator is a difference of squares, which factors to $x +5 x -5$ . Cancel the common factor $x +5$ to get the simplified function $\frac{1}{x -5}$ .

It is important to find the domain before simplifying a rational function because simplifying can sometimes remove factors that indicate domain restrictions. For example, in the function $\frac{x +5}{x^{2} -25}$ , simplifying to $\frac{1}{x -5}$ removes the factor $x +5$ , which indicates that $x =-5$ is a restriction. If you simplify first, you might miss this restriction, leading to an incorrect domain.

Common mistakes when working with rational functions include: 1) Not finding the domain before simplifying, which can lead to missing domain restrictions. 2) Incorrectly factoring polynomials in the numerator or denominator. 3) Forgetting to cancel common factors completely. 4) Assuming the simplified function has the same domain as the original without checking. Always ensure you find the domain first, factor correctly, and verify the domain after simplification.