Problem 21 - College Algebra

Question

For the following exercises, find the domain of each function using interval notation. $$ f(x)=\frac{2 x+1}{\sqrt{5-x}} $$

Step-by-Step Solution

Answer

The domain of the function is \((-\infty, 5)\).

Step 1: Identify Restriction on the Denominator

The denominator of the function is the square root expression \( \sqrt{5-x} \). Since the square root function is only defined for non-negative numbers, we need \( 5-x \geq 0 \).

Step 2: Solve the Inequality

To solve \( 5-x \geq 0 \), we isolate \( x \) and get \( x \leq 5 \). This means that \( x \) must be less than or equal to 5 for the square root to be defined.

Step 3: Consider the Domain Based on Denominator

Since the denominator can't be zero or negative, the function is undefined if \( \sqrt{5-x} = 0 \) or if it becomes negative. We must have \( 5-x > 0 \), or equivalently \( x < 5 \), eliminating \( x = 5 \) because it would make the denominator zero.

Step 4: Write Domain in Interval Notation

Based on the analysis, the domain of \( f(x) \) is all real numbers \( x \) that satisfy \( x < 5 \). In interval notation, this is represented as \((-\infty, 5)\).

Key Concepts

Interval Notation

Interval notation provides a concise way to express the set of numbers that form the domain of a function. It uses brackets to denote closed intervals and parentheses for open intervals.
For example:

• Closed interval: The set \([a, b]\) includes all numbers between \(a\) and \(b\), including \(a\) and \(b\) themselves. This is used when a function is defined at the endpoints.
• Open interval: The set \((a, b)\) includes all numbers between \(a\) and \(b\), but not \(a\) and \(b\) themselves. This is suitable when the function cannot include these boundary values.
• Semi-open interval: \([a, b)\) or \((a, b]\) indicates one endpoint is included, while the other is not.

In our exercise, the domain of the function is expressed as \((-\infty, 5)\). This symbolizes that the function can take any value of \(x\) less than 5, but not equal to 5.

Square Root Function

The square root function is a common mathematical function represented by the symbol \(\sqrt{\cdot}\). It requires special attention because it is only defined for non-negative numbers.
This means that the expression inside the square root, called the radicand, must be zero or positive. For example, the function \(\sqrt{x}\) is only defined when \(x \geq 0\).

• If \(x = 0\), then the square root is zero, which is valid.
• If \(x > 0\), then the square root returns a positive number, which is also valid.
• But if \(x < 0\), the square root leads to an imaginary number in basic real number arithmetic, meaning it is undefined in this context.

In the original exercise, the function's denominator, \(\sqrt{5-x}\), indicates that \(5-x\) should be non-negative for the function to have real number outputs.

Inequalities in Domain Determination

Inequalities play a crucial role when determining the domain of functions, especially when restrictions are imposed by expressions under a square root or in denominators.
To find when a function like \(\frac{2x+1}{\sqrt{5-x}}\) is defined, consider when the denominator is valid. The expression \(\sqrt{5-x}\) specifies that \(5-x\geq 0\). Solving this inequality by isolating \(x\) gives \(x \leq 5\).
However, further reasoning tells us that the square root must not be zero in a denominator, leading to \(\sqrt{5-x} > 0\) or \(5-x > 0\), simplifying to \(x < 5\). This adjustment ensures the expression does not divide by zero. Thus, inequalities help define where a function is valid by identifying constraints imposed by denominators and square roots, determining a more restrictive domain, which in turn eliminates zero from denominators and ensures the radicand's non-negativity.