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Chapter 5: Problem 16

Find the domain of each rational function. $$ R(x)=\frac{5 x^{2}}{3+x} $$

Short Answer

Expert verified
The domain of \( R(x) = \frac{5x^2}{3+x} \) is \(( -\infty, -3 ) \cup ( -3, \infty ) \).

Step by step solution

01

Identify the Rational Function

Given the rational function \[ R(x) = \frac{5x^2}{3+x} \]. The function has a numerator 5x^2 and a denominator 3+x.
02

State the Rule for Domain

The domain of a rational function is all real numbers except where the denominator is zero because division by zero is undefined.
03

Set Up the Denominator

Set the denominator equal to zero and solve for x: \[ 3 + x = 0 \].
04

Solve for x

Solving the equation \[ 3 + x = 0 \] results in \[ x = -3 \].
05

Determine the Domain

The function is defined for all real numbers except x = -3. Hence, the domain of the function is all real numbers except for -3.
06

State the Domain

In interval notation, the domain can be written as: \[ (-\infty, -3) \cup (-3, \infty) \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

rational functions
A rational function is a function represented by a fraction where both the numerator and the denominator are polynomials. In this type of function, the numerator and denominator can have any degree (power of x). For example, in the function \( R(x) = \frac{5x^2}{3+x} \), 5x^2 is the numerator and 3+x is the denominator.
Rational functions are interesting because their domains are not always all real numbers. This is because division by zero is undefined, meaning we need to exclude any values of x that make the denominator zero. Understanding rational functions is key to solving many equations and finding their domains.
domain of a function
The domain of a function is the set of all possible input values (x-values) that make the function work without any mathematical errors. For rational functions, we particularly look at the denominator because dividing by zero is not allowed.
Here's a step-by-step approach to finding the domain of the rational function \( R(x) = \frac{5x^2}{3+x} \):
- Identify where the denominator equals zero: Set 3 + x = 0.
- Solve for x: Here \( x = -3 \).
- Exclude this x-value from the domain.
So, the domain includes all real numbers except x = -3, ensuring that the function remains defined and valid.
interval notation
Interval notation is a way of writing subsets of the real number line. It's concise and clearly shows which numbers are included in the domain.
For the function \( R(x) = \frac{5x^2}{3+x} \), we found that x = -3 makes the denominator zero. This means the domain is all real numbers except -3.
Here's how we write this in interval notation:
- \( (-fty, -3)\) means all numbers less than -3.
- \((-3, fty)\) means all numbers greater than -3.
Putting these together, we get \( (-fty, -3) \cup \ (-3, fty) \), meaning all real numbers except -3. This notation makes it easy to see the range of x-values where the function is defined.

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Most popular questions from this chapter

Use the Rational Zeros Theorem to find all the real zeros of each polynomial function. Use the zeros to factor \(f\) over the real numbers. $$ f(x)=x^{4}-x^{3}-6 x^{2}+4 x+8 $$

Use the Rational Zeros Theorem to find all the real zeros of each polynomial function. Use the zeros to factor \(f\) over the real numbers. $$ f(x)=2 x^{4}+x^{3}-7 x^{2}-3 x+3 $$

Solve each equation in the real number system. $$ 2 x^{3}+3 x^{2}+2 x+3=0 $$

Solve each equation in the real number system. $$ 2 x^{4}-19 x^{3}+57 x^{2}-64 x+20=0 $$

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the vertex of the graph of \(f(x)=3 x^{2}-12 x+7\)
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