Problem 63 - College Algebra and Calculus: An Applied Approach

Question

Find the domain of the function. \(f(x)=\frac{\sqrt{x+1}}{x-2}\)

Step-by-Step Solution

Answer

The domain of the function \(f(x)=\frac{\sqrt{x+1}}{x-2}\) is \(x \in [-1, 2) \cup (2, +\infty)\)

Step 1: Setting the denominator not equal to zero

Firstly, the denominator of the fraction must not be equal to zero, since division by zero is undefined. This leads to the inequality \(x-2 \neq 0\), which gives \(x \neq 2\). So the first part of the domain consists of all real numbers except 2.

Step 2: Setting the value under square root greater than or equal to zero

Next, the value under the square root must be greater than or equal to zero because the square root of a negative number is undefined in the real number system. This leads to the inequality \(x+1 \geq 0\), which gives \(x \geq -1\). So the second part of our domain consists of all real numbers greater than or equal to -1.

Step 3: Combining both parts

We then combine the first part and the second part of the domain. To do so, we take the intersection of the values obtained from steps 1 and 2, because our x must satisfy both conditions. This gives us all real numbers greater than or equal to -1, but not equal to 2.

Key Concepts

Understanding the Domain of a Function

The domain of a function is the complete set of possible values of the independent variable, usually denoted as x, for which the function is defined. Finding the domain involves identifying all the x-values that will give a real number when substituted into the function. For example, in the function,
\(f(x) = \frac{\sqrt{x+1}}{x-2}\)
, we look at both the square root and the fraction to determine where the function can exist. The numerator, which includes a square root, tells us that \(x+1\) must be greater than or equal to zero since we cannot take the square root of a negative number in the real number system. The denominator, \(x-2\), cannot be zero because division by zero is undefined. Therefore, x cannot be 2. By understanding these rules, we can accurately find the domain which consists of all real numbers \(x \geq -1\) but \(x eq 2\).

Solving Inequalities in Algebra

Inequalities are mathematical expressions involving the symbols '>', '<', '\(\geq\)', or '\(\leq\)' to indicate the relative size of two values. Solving inequalities is essential when determining the domain of a function. In our example, we encounter two inequalities: \(x - 2 eq 0\) and \(x + 1 \geq 0\).

The first inequality excludes a single point, which in our case is \(x = 2\), while the second inequality informs us that \(x\) has to be greater than or equal to \(-1\). When combined, these allow us to graph the possible values for \(x\), creating a visual representation which often helps to better grasp the set of admissible values. By understanding how to solve and represent inequalities, students become equipped to handle more complex functions and their domains.

Square Root Functions and Their Domains

Square root functions are a subset of radical functions, with the basic form \(f(x) = \sqrt{x}\). When finding the domain for these functions, the argument under the square root sign must be non-negative, since the square root of a negative number is not a real number.

In our example function, \(f(x) = \frac{\sqrt{x+1}}{x-2}\), the presence of the square root requires that \(x+1 \geq 0\), restricting x to values \(x \geq -1\). Understanding the behavior of square root functions, especially their domains, is crucial in graphing and solving equations that involve square roots. Learning how these functions work provides a foundation for tackling other types of radical functions and more complex mathematical concepts.