Introduction to Rational Functions: Videos & Practice Problems

Rational functions are formed by placing polynomials in a fraction, represented as $p\left(x\right)$ divided by $q\left(x\right)$. The domain is restricted where the denominator equals zero, requiring careful evaluation. To simplify rational functions, factor both the numerator and denominator, canceling common factors. Always determine the domain before simplification to avoid missing restrictions. Understanding these concepts is crucial for working with rational equations effectively.

Intro to Rational Functions

Rational functions are formed by taking a polynomial in the numerator and another polynomial in the denominator, expressed as \( \frac{p(x)}{q(x)} \). Here, \( p(x) \) and \( q(x) \) can be polynomials of any degree. A key aspect of rational functions is that the denominator cannot equal zero, as this would make the function undefined. This restriction leads us to define the domain of the function, which consists of all real numbers except those that make the denominator zero.

To find the domain of a rational function, set the denominator equal to zero and solve for \( x \). For example, for the function \( \frac{3}{3x - 12} \), 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 the function \( f(x) = \frac{x + 5}{x^2 - 25} \), we set the denominator equal to zero:

\[ x^2 - 25 = 0 \]

This factors to \( (x - 5)(x + 5) = 0 \), giving us \( x = 5 \) and \( x = -5 \) as restrictions. Thus, the domain is all real numbers except \( x = 5 \) and \( x = -5 \).

Another important concept 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 \( \frac{3}{3x + 12} \), the denominator can be factored as \( 3(x + 4) \). Canceling the common factor of 3 results in the simplified function \( \frac{1}{x + 4} \).

In the case of \( f(x) = \frac{x + 5}{x^2 - 25} \), the denominator factors to \( (x + 5)(x - 5) \). Canceling the common factor \( x + 5 \) leads to the simplified function \( \frac{1}{x - 5} \). However, it is crucial to determine the domain before simplifying, as simplification may obscure additional restrictions.

In summary, understanding the domain and the process of simplifying rational functions is essential for working effectively with these mathematical expressions. Always ensure to identify any restrictions on the variable before proceeding with simplification to avoid overlooking critical values that affect the function's validity.

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

• A. {x∣x≠0}, $f\left(x\right)=\frac{1}{x-3}$
• B. {x∣x≠3}, $f\left(x\right)=\frac{x^2+9}{x-3}$
• C. {x∣x≠−3}, $f\left(x\right)=\frac{x^2+9}{x-3}$
• D. {x∣x≠3}, $f\left(x\right)=x+3$

Answer: (B)

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

• A. ${\left\lbrace x|x\ne2,-2\right\rbrace},f(x)=\frac{3x^5}{x^2-4}$
• B. ${\left\lbrace x|x\ne2,-2\right\rbrace},f(x)=\frac{6x^5}{2x^2-8}$
• C. ${\left\lbrace x|x\ne2\right\rbrace},f(x)=\frac{3x^5}{x^2-4}$
• D. ${\left\lbrace x|x\ne2\right\rbrace},f(x)=\frac{3x^5}{x^2-8}$

Answer: (A)

Here’s what students ask on this topic:

What is a rational function and how is it different from a polynomial function?

A rational function is a function that can be expressed as the ratio of two polynomials, written as $\frac{p}{q}$, where $p$ and $q$ are polynomials. The key difference between a rational function and a polynomial function is that a polynomial function is a single polynomial expression, while a rational function involves a fraction with a polynomial in both the numerator and the denominator. This introduces domain restrictions where the denominator equals zero, which is not a concern in polynomial functions.

How do you find the domain of a rational function?

To find the domain of a rational function, you need to determine where the denominator is equal to zero, as these values are not allowed in the domain. Set the denominator equal to zero and solve for $x$. The domain will be all real numbers except these solutions. For example, for the function $\frac{1}{x-1}$, set $x-1=0$ and solve to get $x=1$. Thus, the domain is all real numbers except $x=1$.

How do you simplify a rational function?

To simplify a rational function, factor both the numerator and the denominator to identify and cancel out any common factors. For example, consider the function $\frac{x+5}{x^2-25}$. Factor the denominator as $(x+5)(x-5)$. The numerator remains $x+5$. Cancel the common factor $x+5$ to get $\frac{1}{x-5}$. Always determine the domain before simplifying to avoid missing restrictions.

Why is it important to find the domain before simplifying a rational function?

It is crucial to find the domain before simplifying a rational function because simplifying can sometimes eliminate factors that indicate domain restrictions. If you simplify first, you might miss these restrictions. For example, in the function $\frac{x+5}{x^2-25}$, simplifying to $\frac{1}{x-5}$ might make you overlook that $x=-5$ is also a restriction. Always determine the domain first to ensure all restrictions are accounted for.

What are common mistakes to avoid when working with rational functions?

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 the numerator or denominator, which can result in incorrect simplification. 3) Forgetting to cancel common factors completely. 4) Misinterpreting the domain restrictions, especially when dealing with complex fractions. Always carefully factor, cancel common factors, and determine the domain before and after simplification to avoid these errors.