Problem 12 - Algebra and Trigonometry

Question

An expression is given. (a) Evaluate it at the given value. (b) Find its domain. $$ \frac{1}{\sqrt{x-1}}, \quad x=5 $$

Step-by-Step Solution

Answer

Evaluated at \( x=5 \), the expression is \( \frac{1}{2} \). The domain is \((1, \infty)\).

Step 1: Substitute the given value

Substitute the value \( x = 5 \) into the expression \( \frac{1}{\sqrt{x-1}} \). This yields: \( \frac{1}{\sqrt{5-1}} \).

Step 2: Simplify the expression

Calculate the expression \( \sqrt{5-1} \). This simplifies to \( \sqrt{4} = 2 \). Thus, the expression becomes \( \frac{1}{2} \).

Step 3: Evaluate the expression

The evaluated expression with \( x = 5 \) is \( \frac{1}{2} \).

Step 4: Determine the domain

The expression \( \frac{1}{\sqrt{x-1}} \) requires the denominator \( \sqrt{x-1} eq 0 \), which means \( x-1 > 0 \). Therefore, the constraint is \( x > 1 \).

Step 5: Write the domain in interval notation

The domain of the expression, based on the constraint \( x > 1 \), is written in interval notation as \((1, \infty)\).

Key Concepts

Domain of a Function

In mathematics, the domain of a function refers to all the possible input values (usually represented as "x") that make the function work without any mathematical errors. It's like the set of legal moves in a game. Every valid input has to make sense for the function, ensuring it doesn't break any mathematical rules, like division by zero or taking a square root of a negative number.

For the expression \( \frac{1}{\sqrt{x-1}} \), finding the domain involves identifying all the x-values where the expression is defined and valid. Since the denominator cannot be zero, the term \( \sqrt{x-1} \) must be greater than zero, because square roots of negative numbers are not real, and division by zero is undefined.

• This condition translates to \( x-1 \gt 0 \), implying \( x \gt 1 \).

Consequently, the domain of this expression, where it operates without issue, is all real numbers greater than 1, usually expressed in interval notation as \((1, \infty)\). This tells us, in the language of mathematics, that \( x \) can take any value greater than 1 and up to infinity, ensuring the expression remains valid.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, etc. These expressions can sometimes cause confusion, especially when dealing with negative values or fractions. Understanding radicals is essential for simplifying and evaluating complex expressions.

In the given expression \( \frac{1}{\sqrt{x-1}} \), we encounter a square root in the denominator. The process requires careful handling because:

• If the number under the square root \( (x-1) \) is negative, the square root becomes imaginary, leading to complications if you're restricted to real numbers.
• The expression is undefined when the square root equals zero because it results in division by zero, which is mathematically unacceptable.

To handle radical expressions correctly, calculate the root first, while ensuring the value under the root is non-negative and non-zero. For instance, when evaluating \( \sqrt{5-1} \), simplify to \( \sqrt{4} \), which is 2. Ensure your calculations adhere to these checks to maintain validity.

Evaluation of Expressions

Evaluating an expression means finding its value based on specific inputs. This process is fundamental in mathematics as it transforms abstract formulas into concrete numbers through substitution and simplification.

Consider the expression \( \frac{1}{\sqrt{x-1}} \) at \( x=5 \). Here’s how it is evaluated step by step:

• Start by substituting the value \( x = 5 \) into the expression, yielding \( \frac{1}{\sqrt{5-1}} \).
• Next, simplify the term under the square root: \( \sqrt{5-1} \) simplifies to \( \sqrt{4} \), which equals 2. This step reveals the power of simplification, reducing complex expressions to their core form.
• Finally, divide 1 by the result of the square root, resulting in \( \frac{1}{2} \).

Evaluation confirms how expressions can be systematically simplified, aligning with defined mathematical principles. By following similar steps, any expression can be accurately evaluated, revealing its true value for any given input.