Problem 21 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify Constraints

To find the domain of the function \( \frac{2x+1}{\sqrt{5-x}} \), we first find any restrictions on \( x \). Since the denominator involves a square root, the expression under the square root, \( 5-x \), must be greater than zero.

Step 2: Solve Inequality

We solve the inequality \( 5 - x > 0 \). Rearranging this, we get \( x < 5 \). This tells us \( x \) must be less than 5 for the function to be defined.

Step 3: Determine Domain Using Interval Notation

Since the only restriction is \( x < 5 \), the domain of the function in interval notation is \( (-\infty, 5) \).

Key Concepts

Interval Notation

Interval notation is a concise way to express the set of all possible input values, or domain, for which a function is defined. It uses brackets and parentheses to describe the domain range. In the exercise, the domain is expressed as \((-\infty, 5)\).

Here, the round parenthesis \(()\) means that 5 is not included in the domain. This is because the function \(\frac{2x+1}{\sqrt{5-x}}\) cannot have 5 as a value for \(x\) due to its restriction. If you see a square bracket \([]\), it includes the endpoint in the domain.

Here's a quick guide for understanding interval notation:

• \((a, b)\): All numbers between \(a\) and \(b\), but not including \(a\) and \(b\).
• \([a, b]\): All numbers between \(a\) and \(b\), including \(a\) and \(b\).
• Use \(-\infty\) or \(\infty\) when the domain is unbounded in one direction.
• \((-\infty, a)\): All numbers less than \(a\).
• \([a, \infty)\): All numbers greater than or equal to \(a\).

Understanding these notations helps in quickly identifying which values \(x\) can take for a function to work without errors, such as division by zero or negative square roots.

Inequality Solving

Inequality solving is crucial in finding the domain of functions, especially where restrictions are due to square roots or fractions. Here, our focus is on solving the inequality from the given function \(\frac{2x+1}{\sqrt{5-x}}\).

Let's break down how to solve the inequality \(5-x > 0\):

• First, move \(x\) to the other side: \(5 > x\).
• This can also be written as \(x < 5\).
• This inequality shows the values \(x\) can take before making the denominator zero or negative, which are not allowed since the square root is involved.

Solving inequalities allows us to define the boundaries within which a function remains valid. This ensures no mathematical mistakes, such as attempting to divide by zero or evaluating the square root of a negative number, occur.

Practicing inequality solving improves your ability to find these restrictions quickly and confidently. This process is foundational in determining the domain of many functions.

Restrictions on Variables

Understanding restrictions on variables is crucial for correctly determining a function's domain. Such restrictions ensure the function operates correctly without running into undefined operations or errors.

For the function \(\frac{2x+1}{\sqrt{5-x}}\), the restrictions arise from the square root in the denominator. Consider these key restrictions:

• Values under the square root must always be greater than zero (for real numbers).
• This prevents \(x = 5\) because it causes \(5-x\) to become zero, leading to a division by zero.
• If \(x > 5\), the expression under the square root becomes negative, and the square root of a negative number is not real.

Identifying these restrictions is part of understanding the limitations of mathematical functions concerning their variables. These restrictions shape the domain and ensure every input into the function is valid and executable.

Recognizing such constraints and their impacts is vital to maintain consistency within your computations, prevent errors, and allow for accurate function analysis.