Problem 4 - Precalculus : Building Concepts and Connections

Question

What is the domain of the function \(f(x)=\sqrt{x-1} ?\)

Step-by-Step Solution

Answer

The domain of the function \(f(x)=\sqrt{x-1}\) is \(x: [1, +\infty)\)

Step 1: Set the Radicand Greater or Equal to Zero

First of all, set the radicand, which is the expression under the square root, greater or equal to zero. So, you will have \(x-1 \geq 0\).

Step 2: Solve the Resulting Inequality

Next move is to solve the resulting inequality for \(x\). By adding 1 to both sides of the equation, you get \(x \geq 1\). That means \(x\) is greater than or equal to 1.

Step 3: Write Down the Domain of the Function

The last step is to express the domain of the function \(f(x)\). In this case, the domain of \(f(x)=\sqrt{x-1}\) is all real numbers greater than or equal \to 1. Or, in interval notation as \(x: [1, +\infty)\).

Key Concepts

Square Root Function

A square root function is a type of function that involves the mathematical operation square root, denoted by the symbol \( \sqrt{ } \). The function \( f(x) = \sqrt{x} \) serves as the most fundamental form of a square root function. It essentially asks the question: what number, when multiplied by itself, equals \( x \)?
It's crucial to remember that the square root function is only defined for non-negative values because you cannot take the square root of a negative number and obtain a real number result. This restriction affects the domain of the function. For example, in \( f(x) = \sqrt{x-1} \), the expression under the square root is \( x-1 \).

• The inside of the square root (radicand) must be zero or positive.

To determine if a square root function is defined, set the radicand \( x-1 \) greater than or equal to zero, forming an inequality that leads to determining the domain.

Inequalities

Inequalities involve expressions that utilize inequality symbols like \( >, <, \geq, \leq \). These symbols are critical when discussing functions like square root functions as they help define valid input (domain) values for functions.
By using inequalities, you ensure that the condition for which the function exists, for example in \( f(x) = \sqrt{x-1} \), is satisfied.
When determining where a square root is defined, you typically set up an inequality such as \( x-1 \geq 0 \). Solving this inequality provides the values of \( x \) for which the function is valid:

• First, express the condition as an inequality.
• Then, solve it to find the domain of the function.

For the function \( f(x) = \sqrt{x-1} \), when you solve the inequality \( x-1 \geq 0 \), you add 1 to both sides to get \( x \geq 1 \). This confirms the domain requirement for the square root.

Interval Notation

Interval notation is a mathematical notation used to represent the set of values that are solutions to an inequality. It provides a concise way to describe the domain of a function, hinting at what values \( x \) can take. It's commonly used in calculus to denote intervals on the real number line.
For an interval, the notation includes:

• Brackets: [ ] for inclusive bounds, meaning the number is part of the interval.
• Parentheses: ( ) for exclusive bounds, meaning the number is not part of the interval.
• Arrows such as \( \infty \) or \( -\infty \) to indicate continuing indefinitely.

In the problem \( f(x) = \sqrt{x-1} \), where the domain is \( x \geq 1 \), the interval notation is written as \([1, +\infty)\). This tells us \( x \) starts at 1 and extends to infinity, including 1, but not infinity itself. This notation simplifies and clarifies the domain of functions dramatically.