Problem 16 - College Algebra

Question

For the following exercises, find the domain of each function using interval notation. $$ f(x)=\frac{\sqrt{x+4}}{x-4} $$

Step-by-Step Solution

Answer

Domain: \([-4, 4) \cup (4, \infty)\)

Step 1: Recognize Function Type and Constraints

The function \( f(x) = \frac{\sqrt{x+4}}{x-4} \) contains a square root in the numerator and a linear expression in the denominator. To find the domain, we need to ensure that all parts of the function are defined. This means:1. The expression \( x+4 \) under the square root must be non-negative.2. The denominator \( x-4 \) cannot equal zero.

Step 2: Determine Square Root Constraint

For the square root expression \( \sqrt{x+4} \) to be defined, the inside, \( x+4 \), must be greater than or equal to zero. Solve the inequality:\[ x+4 \geq 0 \]Subtract 4 from both sides:\[ x \geq -4 \]This means \( x \) must be greater than or equal to \(-4\) for the square root to be valid.

Step 3: Determine Denominator Constraint

The denominator must not be zero to ensure the function is defined at all points:\[ x-4 eq 0 \]Solve this by adding 4 to both sides:\[ x eq 4 \]This means \( x \) cannot be 4.

Step 4: Combine Constraints

Combine the results from Steps 2 and 3 to find the domain of the function. The domain is where both conditions are satisfied:- \( x \geq -4 \)- \( x eq 4 \)So the domain of \( f(x) \) in interval notation is \([-4, 4) \cup (4, \infty)\).

Key Concepts

Square Root Function

The square root function is often represented as \( \sqrt{x} \) in algebra. It involves taking the square root of a number or expression. For the square root function to be defined, the expression inside the square root must be non-negative. This is because square roots of negative numbers are not real. Here are some key points:

• The expression inside the square root (radicand) must be zero or positive.
• Determine the constraints on the variable by solving the inequality for the radicand.

In the function \( f(x) = \frac{\sqrt{x+4}}{x-4} \), the square root portion is \( \sqrt{x+4} \). Therefore, we solve \( x+4 \geq 0 \), leading to \( x \geq -4 \). This ensures the radicand is non-negative, so the square root is valid.

Interval Notation

Interval notation is a mathematical way to represent a range of values for which a function is defined. It's a concise method to describe inequalities related to functions. Here are its components:

• Brackets \( [ ] \) indicate the endpoints are included in the interval.
• Parentheses \( ( ) \) imply the endpoints are not included.

For the function given, we sought the domain through constraints. We found that \( x \geq -4 \) and \( x eq 4 \). Therefore, the domain in interval notation is expressed as \([-4, 4) \cup (4, \infty)\). This indicates that \( x \) includes values from \(-4\) to just below \(4\), and values above \(4\) extending to infinity, aligning all conditions.

Rational Function

A rational function is expressed as the quotient of two polynomials. The function \( f(x) = \frac{\sqrt{x+4}}{x-4} \) is also a rational function. When dealing with rational functions:

• Identify any restrictions where the denominator equals zero, as these points are undefined.
• Analyze the numerator for additional constraints (like square roots).

Here, the denominator of \( f(x) \) is \( x-4 \). This implies \( x eq 4 \), as the function would be undefined at this point. Additionally, combining this with the requirement for the square root, we obtain the domain \([-4, 4) \cup (4, \infty)\). This takes into account both the numerator and the denominator's restrictions.