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Chapter 2: Problem 3

In Exercises 1鈥�30, find the domain of each function. $$ g(x)=\frac{3}{x-4} $$

Short Answer

Expert verified
The domain of the function \(g(x)=\frac{3}{x-4}\) is all real numbers, \(x\neq4\), or equivalently \((-\infty, 4) \cup (4, +\infty)\).

Step by step solution

01

Set the denominator equal to zero

The denominator of the function is \(x-4\). By setting it equal to zero, we obtain the equation \(x-4=0\). The solutions to this equation are the x-values that make the function undefined.
02

Solve the equation

If we solve the equation \(x-4=0\) for \(x\), we get \(x=4\). This is the x-value that makes the denominator, and consequently the function, undefined.
03

Exclude the undefined x-value from the domain

Now that we know what x-value makes the function undefined, we can describe the domain of the function. All real number except \(x=4\) are part of the domain of the function. We write this as \(x\neq4\) or in interval notation as \((-\infty, 4) \cup (4, +\infty)\).

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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 expressions involving divisions where both the numerator and the denominator are polynomials. An example is the function \(g(x) = \frac{3}{x-4}\). Here, \(3\) is the numerator and \(x-4\) is the denominator. These types of functions are significant due to their characteristics linked with values that make the denominator zero. Be careful with rational functions because they can lead to division by zero, which makes them undefined at certain points. Understanding these functions requires identifying such points and ensuring they are excluded from the domain.
Undefined Values
In rational functions, undefined values occur when the denominator equals zero. This is because division by zero is not permissible in mathematics. For instance, with the function \(g(x) = \frac{3}{x-4}\), setting the denominator \(x-4\) to zero reveals that \(x=4\) is the undefined value. This specific x-value disrupts the function, as the division by zero occurs, causing the function to be non-existent at this point. When determining the domain of a function, it is critical to identify and exclude these undefined values.
Interval Notation
Interval notation is a concise way to describe sets of numbers, frequently used to denote the domain of a function. It utilizes brackets and parentheses to indicate whether endpoints are included or excluded from the set. For domains of rational functions, such as \(g(x) = \frac{3}{x-4}\), we exclude the undefined value from \(x = 4\). We express this in interval notation as \((-fty, 4) \cup (4, +fty)\). Here, parentheses mean 4 is not included, showing clearly which intervals are part of the domain.
Real Numbers
Real numbers include all the numbers on the continuous number line, covering both rational and irrational numbers. They include integers, fractions, and transcendental numbers like \(\pi\) and \(e\). When determining the domain of a rational function, we aim to include all real numbers except those that make the function undefined. For \(g(x) = \frac{3}{x-4}\), every number is part of the domain except \(x = 4\). The category of real numbers forms the backbone of function domains in calculus and algebra.

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