Problem 16 - College Algebra with Corequisite Support

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: Analyze the function

The given function is \( f(x) = \frac{\sqrt{x+4}}{x-4} \). The numerator is \( \sqrt{x+4} \) and the denominator is \( x-4 \). We need to determine where this function is defined.

Step 2: Determine the domain of the numerator

The domain of \( \sqrt{x+4} \) includes all \( x \) such that the expression inside the square root is non-negative. Thus, \( x+4 \geq 0 \), which simplifies to \( x \geq -4 \). So, \( x \) must be greater than or equal to \(-4\).

Step 3: Determine the domain restriction from the denominator

The function is undefined when the denominator is zero. So, we need \( x - 4 eq 0 \), which simplifies to \( x eq 4 \). Thus, \( x \) cannot be \(4\).

Step 4: Combine restrictions to find the domain

Considering both the square root and the denominator's restriction, the domain of the function is all \( x \) such that \( x \geq -4 \) and \( x eq 4 \). In interval notation, this is \([ -4, 4 ) \cup ( 4, \infty ) \).

Key Concepts

Interval Notation

Interval notation is a mathematical method for describing a range of values. It is often used to represent domains of functions, which are the set of all possible input values. An interval can be closed, open, or a combination of both, inclusive or exclusive of endpoints.

Here's how it works:

• Closed Interval: Denoted by square brackets [a, b], it includes all numbers between a and b, including both endpoints.
• Open Interval: Denoted by parentheses (a, b), it includes numbers between a and b, but not the endpoints themselves.
• Half-Open Interval: This is a combination with one endpoint included and the other not, such as [a, b) or (a, b].

Now, when we talk about the domain of a function like \(f(x)=\frac{\sqrt{x+4}}{x-4}\), the goal is to identify all the x-values that make the function work without any issues like division by zero or negative numbers inside a square root. The usage of interval notation in the step-by-step solution helps clearly convey which x-values are permissible, rendering the expression error-free. Therefore, for our given function, using previous calculations, the domain is described by the interval notation \( [ -4, 4 ) \cup ( 4, \infty ) \). This means all numbers greater than or equal to -4, excluding 4 itself. This style concisely conveys the valid input range for the function.

Radical Expressions

Radical expressions include the root of a number or expression, such as the square root used in \(\sqrt{x+4}\). This type of expression is integral for understanding the possible domains of a function since there are often specific restrictions due to the nature of radicals.

For a square root, the expression inside – known as the radicand – should never be negative if we are dealing only with real numbers. This is because the square root of a negative number results in an imaginary number. So, for \(\sqrt{x+4}\) to be real, we need \(x+4 \geq 0\), simplifying to \(x \geq -4\). This tells us that only x-values from -4 and greater will keep the output within the realm of real numbers. Taking special note of these restrictions is crucial. It ensures that the function provides usable, real-number outputs.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. It's important because solving or determining the domain of these expressions requires careful analysis to avoid division by zero.

In the function \(f(x)=\frac{\sqrt{x+4}}{x-4}\), the denominator is \(x-4\). If this bottom part equals zero, the expression becomes undefined because you cannot divide by zero. Therefore, we must ensure \(x-4 eq 0\), leading us to conclude \(x eq 4\). This condition gives us a critical restriction on the domain.

When combined with the domain from the radical expression, we get a complete domain \([ -4, 4 ) \cup ( 4, \infty ) \). Such considerations allow us to fully understand and define where the function is valid, avoiding pitfalls like undefined values.