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Chapter 15: Problem 63

Determine the domain of each function. \(f(x)=\frac{x}{x-3}\)

Short Answer

Expert verified
The domain is \[ (-\infty, 3) \cup (3, \infty) \].

Step by step solution

01

Identify the function

The given function is a rational function: \[ f(x) = \frac{x}{x-3} \]
02

Find the denominator

Notice that the denominator of the function is \(x-3\).
03

Determine the restriction

For the function to be defined, the denominator cannot be zero. Set the denominator equal to zero and solve for \(x\):\[ x - 3 = 0 \] Solving this equation gives: \[ x = 3 \]
04

Write the domain

The domain of the function is all real numbers except the value that makes the denominator zero. Therefore, the domain is: \[ (-\infty, 3) \cup (3, \infty) \]

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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 any function that can be written as the quotient of two polynomials. The general form of a rational function looks like this: \( f(x) = \frac{P(x)}{Q(x)} \), where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x)\) 鈮� 0.
These functions are useful for describing a wide variety of behaviors and phenomena in mathematics.
Because they involve division, we must always pay special attention to the denominator. If the denominator equals zero, the function is undefined.
In many real-world applications, rational functions model scenarios like rate problems, growth models, and more.
Domain Restrictions
The domain of a function refers to all the possible input values (x-values) for which the function is defined.
For rational functions, the most critical step is identifying where the function might be undefined.
  • Look at the denominator of the rational function.
  • Set the denominator equal to zero and solve for \(x\): this gives the values that are not allowed in the domain.
For example, in the function \( f(x) = \frac{x}{x-3} \), the denominator is \( x-3 \).
Setting \( x-3 = 0 \) and solving for \( x \), we find that \( x = 3 \).
Therefore, the function is undefined at \( x = 3 \), and the domain would be all real numbers except \( x = 3 \).
Solving Equations
Solving equations is a fundamental skill in algebra.
When determining the domain of a rational function, you'll need to solve simple equations to find domain restrictions. Here is a step-by-step process:
  • Identify the equation from the denominator of the rational function.
  • Set this equation to zero.
  • Solve for the variable (often \(x\)).
As an illustration, for \( f(x) = \frac{x}{x-3} \), the denominator is \( x-3 \). Setting this equal to zero: \( x-3 = 0 \).
Solving for \( x \), we get \( x = 3 \). This tells us that \( x = 3 \) is the restriction. Hence, the domain is all real numbers except \( x = 3 \).

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