Question

A student says that the domain of \(f(x)=\frac{\sqrt{x+1}}{x-3}\) is all real numbers except \(x=3 .\) Is the student correct? Explain.

Short answer

The student is not correct. The domain of the function \(f(x)=\frac{\sqrt{x+1}}{x-3}\) is \([-1, 3) \cup (3, \infty)\), which includes all real numbers greater than or equal to -1, excluding 3.

Step-by-step solution

Identify the Domain of the Numerator \(\sqrt{x+1}\)

For \(\sqrt{x+1}\), we know that the square root function only takes non-negative values, thus \(x+1\) must be \(\geq 0\). Solving this inequality, we find \(x \geq -1\). So, the domain of the numerator is \([-1, \infty)\).

Identify the Domain of the Denominator \(x-3\)

For the denominator, a fraction cannot have a denominator of zero as it would make the function undefined. So, \(x-3\) cannot equal zero. Solving \(x-3\neq0\), we get \(x \neq 3\). So, the domain of the denominator is \((-\infty, 3) \cup (3, \infty)\).

Combine the two Domains

To combine the two domains, we find the intersection of \([-1, \infty)\) and \((-\infty, 3) \cup (3, \infty)\). In this case, we can see that the function is defined for all real numbers greater than or equal to -1 and not equal to 3. So, the combined domain is \([-1, 3) \cup (3, \infty)\).

Key concepts

Rational Functions

A rational function is a type of function that can be expressed as the quotient of two polynomials, where the numerator and denominator are both polynomial expressions. In mathematical terms, a rational function has the form \(f(x) = \frac{p(x)}{q(x)}\), where \(p(x)\) and \(q(x)\) are polynomials and \(q(x) \eq 0\).

When determining the domain of a rational function, it is crucial to ensure that the denominator \(q(x)\) does not equal zero, since division by zero is undefined in mathematics. The domain of a rational function is therefore all real numbers for which the denominator is non-zero. Besides this restriction, the domain may also be affected by any further restrictions required by the numerator, particularly if it includes a square root or another operation that imposes additional domain constraints.

Inequalities

In mathematics, inequalities are used to compare the size or order of two expressions. They take forms such as \(x > a\), \(x < b\), \(x \geq c\), and \(x \leq d\), where \(|a|, |b|, |c|\), and \(|d|\) are real numbers or expressions. Solving inequalities is integral when finding the domain of functions, particularly square root functions and rational functions, as it helps to determine the range of values for which the function is well-defined and real-valued.

For instance, when you have \(\sqrt{x+1}\), we must solve the inequality \(x+1 \geq 0\) to find the values of \(x\) that make the expression under the square root non-negative. Understanding how to solve inequalities is therefore essential for correctly determining function domains.

Square Root Functions

Square root functions are a specific type of function where the variable \(x\) appears under a square root sign. The general form of a square root function is \(f(x) = \sqrt{g(x)}\), where \(g(x)\) is an algebraic expression. These functions impose a very specific restriction on their domain: the expression under the square root, \(g(x)\), must be greater than or equal to zero, since the square root of a negative number is not a real number.

Therefore, to determine the domain of a square root function, you must first find the set of all \(x\) values for which \(g(x) \geq 0\). This typically involves solving inequalities where the solution set defines the domain of the function, ensuring that the values substituted into the function will yield real numbers.

Function Domain Restrictions

The domain of a function is the set of all possible input values (\(x\)-values) for which the function is defined. In other words, it's the collection of all real numbers that you can plug into the function without encountering any mathematical issues such as division by zero or the square root of a negative number.

When analyzing domain restrictions, we must consider factors such as the existence of denominators in rational functions, which cannot be zero, as well as the expressions under a square root, which must be non-negative. Domain restrictions can often be determined by setting up and solving inequalities, or by finding the intersection of domains when multiple constraints apply. In our exercise, we combined the domain of the square root function \(x \geq -1\) with the domain of the rational function where \(x \eq 3\) to find the function's overall domain.