Problem 20 - College Algebra with Corequisite Support

Question

For the following exercises, find the domain of each function using interval notation. \(\frac{5}{\sqrt{x-3}}\)

Step-by-Step Solution

Answer

The domain is \((3, \infty)\).

Step 1: Identify the restriction

To find the domain of the function \( f(x) = \frac{5}{\sqrt{x-3}} \), we first need to identify the values for which the function is not defined. The function is undefined when the denominator is zero or when the expression inside the square root is negative.

Step 2: Ensure the denominator is not zero

The expression \( \sqrt{x-3} \) must be greater than zero because it is in the denominator. This implies that \( x-3 \) must be strictly greater than zero.

Step 3: Solve the inequality

Solve the inequality \( x-3 > 0 \). This gives \( x > 3 \). This means that \( x \) must be greater than 3 for the function to be defined.

Step 4: Write the domain in interval notation

The domain of the function, considering the restriction \( x > 3 \), is the set of all real numbers greater than 3. In interval notation, this is written as \( (3, \infty) \).

Key Concepts

Interval Notation

Interval notation is a concise way of representing a set of numbers on a number line. By using this notation, we can clearly indicate the start and end points of a range, as well as whether each endpoint is included in the set.

When writing in interval notation, we use brackets:

• "(", ")" indicates that the endpoint is not included (open interval).
• "[", "]" signifies that the endpoint is included (closed interval).

For example, if you have a function where the domain includes all numbers greater than 3, but not 3 itself, it is expressed as \((3, \infty)\). Here, the "(" before 3 shows that 3 is not part of the domain, and "\(\infty\)" means there is no upper limit to the numbers in the domain.

Understanding interval notation helps us easily identify which numbers satisfy the conditions of a function.

Square Root Function

The square root function is represented as \(\sqrt{x}\), which means finding a number that, when squared, gives \(x\). This function is only defined for non-negative numbers because the square root of a negative number is not a real number. This is crucial when determining the domain of functions involving square roots.

When you encounter a square root expression in a function's numerator, denominator, or another location, it is important to ensure the expression inside the square root is zero or positive.

For example, in the function \(f(x) = \frac{5}{\sqrt{x-3}}\), since \(\sqrt{x-3}\) is in the denominator, it must be greater than zero. Therefore, the expression \(x-3\) must be greater than zero, leading to \(x > 3\). This information is vital because it directly influences the domain of the function.

Inequalities

Inequalities are mathematical expressions involving symbols like "<", ">", "≤", or "≥". They show the relationship between two values, indicating how one number is less than, greater than, or equal to another.

Consider an inequality \(x-3 > 0\). Solving it helps establish the values of \(x\) that make the expression true.
To solve \(x-3 > 0\), perform the following step-by-step:

• Add 3 to both sides: \(x > 3\).

This solution tells us all values of \(x\) greater than 3 satisfy the inequality, providing critical information about the domain of functions with constraints.

Inequalities are indispensable tools for describing conditional ranges and expressing solutions in a clear and understandable manner.

Rational Functions

Rational functions are ratios of two polynomial expressions, similar to the way fractions are ratios of numbers. A simple form of rational functions is \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials, and \(Q(x)\) must not be zero because division by zero is undefined.

In the exercise, the rational function \(f(x) = \frac{5}{\sqrt{x-3}}\) means that \(5\) is the numerator and \(\sqrt{x-3}\) is the denominator. For any rational function, it is crucial to identify where the denominator is zero, as these values are not part of the domain.

In this case, since the denominator involves a square root, we also ensure that the square root is positive. Hence, \(x-3 > 0\), or \(x > 3\), which means our function is defined only for \(x > 3\). This highlights the importance of carefully analyzing both the numerator and the denominator to determine valid input values for the function.