Problem 27 - Calculus with Concepts in Calculus

Question

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

Step-by-Step Solution

Answer

The domain of \( g(x) \) is \( (-\infty, 1) \cup (1, \infty) \).

Step 1: Understand the function

The given function is \( g(x) = \frac{2}{x-1} \). This is a rational function, where the denominator cannot be zero.

Step 2: Set the denominator not equal to zero

Identify the expression in the denominator that must not be zero, which is \( x - 1 \). Set up the inequality: \( x - 1 eq 0 \).

Step 3: Solve the inequality

Add 1 to both sides to solve for \( x \). This gives \( x eq 1 \). Thus, \( x \) cannot be equal to 1.

Step 4: Write the domain

Since \( x \) can be any real number except 1, the domain of \( g(x) \) is all real numbers except 1. In interval notation, this is expressed as \( (-\infty, 1) \cup (1, \infty) \).

Key Concepts

Rational Functions

Rational functions are a type of function in mathematics and are expressed as the ratio of two polynomials. For example, the function \( g(x) = \frac{2}{x-1} \) is a rational function because it involves the division of one polynomial by another. The numerator here is the constant 2, and the denominator is \( x - 1 \).
Rational functions often have values that are not defined due to their denominators. When the denominator equals zero, the function becomes undefined because division by zero is not possible in mathematics.

• Always check the denominator for zero values to identify undefined points in a rational function.
• These points are crucial in determining the domain of the function, as they are excluded from the set of possible input values.

Understanding rational functions is key to solving problems related to their domains and behavior.

Inequalities

Inequalities are mathematical statements that show the relationship between expressions that are not equal. In the context of finding the domain of a rational function, inequalities are used to determine which values make the denominator zero.
To solve an inequality, such as \( x - 1 eq 0 \), you need to manipulate the expression to find values of \( x \) that do not satisfy the equality of the denominator to zero. For our example, solving \( x - 1 eq 0 \) involves:

• Recognizing that the solution means \( x \) cannot be 1.
• This is done by isolating \( x \) to find the specific value causing the denominator to be zero.

Through this process, inequalities aid us effectively in excluding specific values from the domain.

Interval Notation

Interval notation is a concise method of describing a set of numbers along a number line. It is used to portray the domain and range of functions succinctly.
In interval notation, parentheses \(( )\) are used to show that an endpoint is not included, while square brackets \([ ]\) indicate inclusion. For the function \( g(x) = \frac{2}{x-1} \), the domain excludes \( x = 1 \), because the function is undefined at that point. Thus, we write the domain as:

• \((-fty, 1)\), meaning \( x \) can be any value less than but not including 1.
• \((1, \fty)\), meaning \( x \) can be any value greater than but not including 1.

By combining these two intervals using a union, we express the domain as \((-fty, 1) \cup (1, \fty)\), succinctly conveying all real numbers except for a single point.