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Chapter 1: Problem 54

Determine the domain of the function. \(f(x)=\sqrt{2 x-6}\)

Short Answer

Expert verified
The domain of the function is \([3, \infty)\).

Step by step solution

01

Identify the function type

The function is a square root function, given by \(f(x)=\sqrt{2 x-6}\). For a square root function, the expression inside the square root must be non-negative.
02

Set up the inequality

Since the expression inside the square root \(2x - 6\) must be non-negative, set up the inequality: \(2x - 6 \geq 0\).
03

Solve the inequality

Solve the inequality \(2x - 6 \geq 0\). Add 6 to both sides, yielding \(2x \geq 6\). Then, divide both sides by 2, resulting in \(x \geq 3\).
04

Determine the domain

The domain consists of all values of \(x\) that satisfy \(x \geq 3\). In interval notation, this is written as [3, \infty).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

square root function
The square root function is a type of function where the variable is inside a square root symbol. A general form is given by \(f(x) = \sqrt{g(x)}\). For the function to be real and defined, the expression inside the square root, g(x), must be non-negative. This means \g(x) \geq 0\. In our example, \(f(x)=\sqrt{2 x-6}\), we need to ensure that \(2x - 6\) is non-negative. This is a key step to determining the function's domain. It informs us about the range of values that x can take to keep the function valid and real.
inequalities in functions
Inequalities play a crucial role in understanding the domain of functions like the square root function. Let's break down how we deal with these inequalities. Since we're dealing with a square root function, we need the expression inside the square root to be greater than or equal to zero. For \(f(x)=\sqrt{2 x-6}\), that translates to \(2x - 6 \geq 0\). Solving for x involves a few steps:
  • Add 6 to both sides to get \2x \geq 6\.
  • Divide both sides by 2, leading to \x \geq 3\.
This inequality tells us that x must be at least 3 for the function to be defined.
interval notation
Interval notation is a compact way of describing the domain of a function. It uses brackets to indicate which numbers are included in the range. For instance, if x must be greater than or equal to 3, we write the domain as \( [3, \infty)\). Here's a quick guide to interpreting interval notation:
  • Square brackets \([ ]\) indicate that the endpoint is included.
  • Parentheses \(()\) indicate that the endpoint is not included.
  • \(\infty\) always uses parentheses because infinity is not a finite number and can't be included.
This notation is both simple and expressive, and it's very useful in mathematics to denote intervals clearly and concisely.

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