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Chapter 3: Problem 6

Find the domain of each rational function. $$f(x)=\frac{-1}{x-2}$$

Short Answer

Expert verified
The domain is all real numbers except \(x = 2\).

Step by step solution

01

Identify the denominator

Look at the denominator of the rational function. The given function is \(f(x) = \frac{-1}{x-2}\). Here, the denominator is \(x-2\).
02

Set the denominator not equal to zero

Rational functions are undefined when the denominator is zero. Set the denominator not equal to zero and solve for \(x\): \[ x-2 eq 0 \]
03

Solve for the variable

Solve the inequality \(x - 2 eq 0\). Add 2 to both sides to get \(x eq 2\).
04

State the domain

The domain of the rational function is all real numbers except the value where the denominator is zero. Thus, the domain is all real numbers except \(x = 2\).

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Key Concepts

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

Rational Functions
Rational functions are an essential concept in algebra. These are functions represented as the ratio of two polynomials. In general, a rational function can be written as: d(x) = \frac{P(x)}{Q(x)} where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x)\) is not zero. Understanding rational functions is crucial because they often appear in various areas of mathematics and science.
Identifying the Denominator
The first step to finding the domain of a rational function is to identify the denominator. The function given in the exercise is: \(f(x) = \frac{-1}{x-2}\). Here, the denominator is \(x-2\). The denominator is important because it tells us the values of \(x\) that might make the function undefined. A function is undefined whenever its denominator equals zero, so always start by identifying and analyzing the denominator.
Solving Inequalities
Once we identify the denominator, the next step is to set up an inequality to find where the denominator is not equal to zero. For example, in the given problem: \(x-2neq 0\) We solve this inequality to find the values of \(x\) that do not make the denominator zero. By solving, we get: \(xneq 2\). This means the function is undefined when \(x = 2\), helping us narrow down the domain of the function.
Domain of a Function
Finally, we state the domain of the rational function based on our previous calculations. The domain includes all real numbers except where the denominator equals zero. For our function: \(f(x) = \frac{-1}{x-2}\), the denominator is zero at \(x = 2\). So, the domain is all real numbers except 2. In set notation, we write this as: \(\text{Domain} = \{ x | xneq 2 \}\). Understanding the domain helps us know where the function is valid and can be used in calculations and other mathematical problems.

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