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Chapter 5: Problem 15

Find the domain of each rational function. $$ R(x)=\frac{4 x}{x-7} $$

Short Answer

Expert verified
The domain is all real numbers except \( x = 7 \), written as \( (-\infty, 7) \cup (7, \infty) \).

Step by step solution

01

Understand the Rational Function

A rational function is a fraction in which both the numerator and the denominator are polynomials. For the given function, \[ R(x) = \frac{4x}{x-7} \] identify the numerator and the denominator.
02

Identify When the Denominator is Zero

The domain of a rational function is all real numbers except those that make the denominator zero. So, find the value of \( x \) that makes the denominator zero. Set the denominator equal to zero and solve for \( x \): \[ x-7 = 0 \] Thus, \( x = 7 \).
03

State the Domain

The domain of the rational function \( R(x) \) consists of all real numbers except the value we found in the previous step. Therefore, the domain is all real numbers except \( x = 7 \). This can be written in interval notation as: \[ (-\infty, 7) \cup (7, \infty) \]

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Key Concepts

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

Rational Function
A rational function is a type of function represented as the ratio of two polynomials. The general form is \[ R(x) = \frac{P(x)}{Q(x)} \] where both \(P(x)\) and \(Q(x)\) are polynomials. In our exercise, \[ R(x) = \frac{4x}{x-7} \] is a rational function because it's written as the fraction of the polynomial \(4x\) in the numerator and \(x-7\) in the denominator.

One key aspect of rational functions is their domains, which are dictated by the values that cannot make the denominator zero.

Understanding the domain is crucial to working effectively with rational functions.
Polynomials
Before diving deeper into rational functions, it's essential to understand polynomials. A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. For example, \(P(x) = 4x\) and \(Q(x) = x-7\) are polynomials.

Polynomials can have various degrees based on the highest power of the variable, but in the context of our exercise, the given polynomials are relatively simple.

Polynomials can be added, subtracted, multiplied, and divided (except by zero), making them versatile in algebra.
Interval Notation
Interval notation is a way of writing subsets of the real number line. For rational functions, we often use interval notation to express the domain.

In the given exercise, we identified the domain excluding the value that makes the denominator zero. This specific domain can be written using interval notation:

\[ (-\frac{\text{{鈭瀩}, 7) \frac{\text{{鈭獇} (7, 鈭瀩) \] which means all real numbers except x = 7.

In interval notation:
  • Parentheses ( ) indicate that the endpoint is not included.
  • Braces [ ] indicate the endpoint is included (not used here).
  • The union symbol 鈭� is used to combine intervals.
When using interval notation, it helps to visualize the number line to understand which values are included.
Denominator Analysis
Analyzing the denominator is a critical step in identifying the domain of a rational function. For the given function:

\[ R(x) = \frac{4x}{x-7} \]

we need to determine which values make the denominator zero.

We solve the equation \(x-7 = 0\) to find \(x = 7\). This means \(x = 7\) must be excluded from the domain since division by zero is undefined.

Hence, the domain of the rational function is all real numbers except \(x = 7\). Understanding which values make the denominator zero and excluding them is key to working with rational functions.

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Most popular questions from this chapter

Solve each equation in the real number system. $$ 2 x^{4}+x^{3}-24 x^{2}+20 x+16=0 $$

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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. Express the inequality \(3 \leq x<8\) using interval notation.
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