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Chapter 10: Problem 93

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

Short Answer

Expert verified
The domain is \[ x \in [6, \infty) \].

Step by step solution

01

- Identify the Function Type

The function given is a square root function: \[ f(x) = \sqrt{x-6} \] Square root functions are only defined for values where the expression inside the square root is non-negative.
02

- Set the Condition for the Square Root

To ensure the value inside the square root is non-negative, set up the inequality: \[ x - 6 \geq 0 \]
03

- Solve the Inequality

Solve the inequality for \(x\): \[ x - 6 \geq 0 \] Adding 6 to both sides gives: \[ x \geq 6 \]
04

- Determine the Domain

From the inequality solution, the domain of the function is all values of \(x\) that are greater than or equal to 6. Therefore, the domain of the function \( f(x) = \sqrt{x-6} \) is: \[ x \in [6, \infty) \]

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Key Concepts

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

Square Root Function
Understanding square root functions is critical for solving problems like the one in the exercise. A square root function has the form \( f(x) = \sqrt{x - a} \), where \(a\) is a constant. The key characteristic of these functions is that 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. For example, in the given problem, the function is \( f(x) = \sqrt{x - 6} \). To ensure we only take the square root of non-negative numbers, we need to set conditions on \(x\) so that \( x - 6 \geq 0 \). This is the foundation for identifying the function's domain.
Inequality
Inequalities are essential tools in determining the domain of functions like the square root function. In the given problem, we need to solve the inequality \( x - 6 \geq 0 \) to find the acceptable values for \(x\). Here's how it works:

- Start by isolating the variable on one side of the inequality.
- For \( x - 6 \geq 0 \), add 6 to both sides to get \( x \geq 6 \).
This means that \(x\) must be greater than or equal to 6. Solving inequalities helps us set boundaries, ensuring the expression inside the square root function remains non-negative. Without this step, we cannot accurately determine the domain.
Domain
The domain of a function is the set of all possible input values (\(x\) values) for which the function is defined. For the given problem, we use the result from our inequality solution to specify the domain. Since we determined \( x \geq 6 \), it means our function \( f(x) = \sqrt{x - 6} \) is only defined when \(x\) is at least 6.

So, the domain of the function can be written as: \( x \in [6, \infty) \). This interval notation denotes all values starting from 6 and extending to infinity, inclusive of 6. Identifying the domain ensures we only consider valid inputs, avoiding undefined scenarios in the function.

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