Problem 45 - Algebra and Trigonometry

Question

Find the domain of the function. $$ f(x)=\sqrt{x-5} $$

Step-by-Step Solution

Answer

The domain is \([5, \infty)\).

Step 1: Understand the Problem

We need to find the domain of the function \( f(x) = \sqrt{x - 5} \). For a function to be defined, the expression inside the square root must be non-negative because the square root of a negative number is not defined in the set of real numbers.

Step 2: Set Up the Inequality

To ensure that the expression under the square root is non-negative, set up the inequality: \( x - 5 \geq 0 \). This inequality ensures that the expression inside the square root will not be negative.

Step 3: Solve the Inequality

Solve the inequality \( x - 5 \geq 0 \). Add 5 to both sides of the inequality to isolate \( x \): \( x \geq 5 \).

Step 4: State the Domain

The solution to the inequality \( x \geq 5 \) gives us the domain of the function. Therefore, the domain of \( f(x) = \sqrt{x - 5} \) is all values of \( x \) such that \( x \geq 5 \). In interval notation, this is \([5, \infty)\).

Key Concepts

Square Root Function

When delving into square root functions, the main idea to understand is the concept of limiting the domain to avoid undefined values. In its essence, a square root function looks like this: \( f(x) = \sqrt{x} \). A distinctive feature of these functions is that the expression under the square root must be non-negative (i.e., zero or positive). This is because square roots of negative numbers are not real but complex.For the function \( f(x) = \sqrt{x-5} \), the domain is the set of all possible values of \( x \) that turn \( x-5 \) into a non-negative number. This means we must solve the inequality \( x - 5 \geq 0 \). The solution to this gives the values of \( x \) where the function will output real numbers.To summarize, for any square root function:

• The expression inside the square root must be \( \geq 0 \).
• The domain is determined by setting the expression \( \geq 0 \) and solving it.

This ensures that the function is defined in the real number system.

Inequalities

Inequalities are mathematical expressions used to compare quantities. When working with square root functions, inequalities help in ensuring the expressions under the square root remain non-negative. In the exercise, we determine the domain by setting an inequality: \( x - 5 \geq 0 \). Here's a simple step-by-step approach to handle this:

• Identify the expression under the square root (here, it's \( x-5 \)).
• Set up an inequality \( x - 5 \geq 0 \) to ensure no negative under the root.
• Solve this inequality. In this instance, adding 5 to both sides yields \( x \geq 5 \).

This inequality has shown us all the \( x \)-values that keep the function defined in real numbers.Remember:

• Solve the inequality like you would any equation while respecting any direction changes when multiplying or dividing by a negative number.
• Always interpret the inequality result to find the domain of the function.

Interval Notation

Interval notation is a concise way to express a range of numbers, often used to denote the domain of a function. Understanding how to use this notation simplifies your ability to communicate domains.For the given function \( f(x) = \sqrt{x-5} \), after solving the inequality \( x \geq 5 \), we express the domain using interval notation. This result shows that \( x \) can be any value starting from 5, extending to infinity. We write this as \([5, \infty)\).
Here's how to interpret and write interval notation effectively:

• \([\) or \(]\) denotes inclusivity of the number. In \([5,\), 5 is included, indicating \( x = 5 \) is valid.
• \(()\) denotes exclusivity, meaning the number is not included. \(\infty)\) implies \( x \) goes indefinitely upward without explicitly including infinity itself.

The use of interval notation streamlines and standardizes how domains, ranges, and solution sets are communicated, avoiding ambiguity.