Problem 10 - Precalculus Mathematics for Calculus

Question

Find the domain of the expression. $$\frac{1}{\sqrt{x-1}}$$

Step-by-Step Solution

Answer

The domain is \( (1, \infty) \).

Step 1: Understanding the Expression

To find the domain of the expression \( \frac{1}{\sqrt{x-1}} \), we first recognize that the expression involves a square root in the denominator. Therefore, for the fraction to be defined, the expression inside the square root must be positive.

Step 2: Setting the Condition for the Square Root

The square root \( \sqrt{x-1} \) is only defined for non-negative numbers. Additionally, since it is in the denominator, it cannot be zero. Therefore, we set up the inequality: \( x-1 > 0 \).

Step 3: Solving the Inequality

Solve the inequality \( x-1 > 0 \) to find the values of \( x \) for which the expression is defined. Adding 1 to both sides, we get \( x > 1 \).

Step 4: Describing the Domain

In interval notation, the solution to the inequality \( x > 1 \) is represented as \( (1, \infty) \). This means that the domain of the expression is all real numbers greater than 1.

Key Concepts

Square Root

The square root is a mathematical operation that finds a number which, when multiplied by itself, results in the given number under the square root sign. The symbol for square root is \(\sqrt{\cdot}\). Understanding square roots is crucial because they frequently appear in problems involving domains, such as in our expression \(\frac{1}{\sqrt{x-1}}\).

• Square roots of negative numbers are not defined in the set of real numbers. Thus, when dealing with square roots, we enforce the rule that the expression inside the root must be non-negative.
• This means that the smallest possible value inside a square root is zero, as it is the boundary where the root still remains defined.

In our problem, \( x-1 \) needs to be positive because it is under a square root sign, and it is also located in the denominator of a fraction.

Inequality Solving

Inequality solving is a critical mathematical skill that helps establish conditions for values. In this problem, we need to solve the inequality \(x-1 > 0\) to determine the domain of our function. Here’s how you approach it:

• Recognize the inequality sign: Inequalities use signs such as \(>\) or \(<\). In our case, \(x-1 > 0\) ensures the expression under the square root is positive.
• Move constants: Solve the inequality \(x-1 > 0\) by isolating \(x\). This can be done by adding 1 to both sides, resulting in \(x > 1\).
• Check the direction: The solution \(x > 1\) tells us which side of the number line our answers will be on. Here, it's all values greater than 1.

Solving inequalities accurately ensures that we're only considering valid \(x\) values that keep the function defined and meaningful.

Interval Notation

Interval notation is a shorthand used in mathematics to express sets of numbers that fall within certain bounds. It is especially helpful for denoting domains of functions. Understanding how it works can greatly enhance reading and writing mathematical solutions.

• An open interval, written as \((a, b)\), includes all numbers between \(a\) and \(b\) but not \(a\) or \(b\) themselves.
• In this problem, \(x > 1\) translates to the interval notation \((1, \infty)\) which indicates all real numbers greater than 1.
• Infinity (\(\infty\)) is always paired with an open bracket, \((\), because it is not a tangible number that we can include in our set.

Using interval notation provides a clear, concise way to describe the domain of functions.