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Chapter 7: Problem 93

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

Short Answer

Expert verified
The domain of the function \( f(x) = \sqrt{x-6} \) is \( [6, \infty) \).

Step by step solution

01

Identify the expression inside the square root

To find the domain of the function, begin by identifying the expression inside the square root. For the function ewline ewline \(f(x)=\sqrt{x-6}\), the expression inside the square root is \(x-6\).
02

Set the expression inside the square root to be non-negative

Since the square root function is only defined for non-negative numbers, set the expression inside the square root to be greater than or equal to zero: \[ x - 6 \geq 0 \].
03

Solve the inequality

Solving the inequality \[ x - 6 \geq 0 \] results in \[ x \geq 6 \].
04

Express the domain in interval notation

Since \(x\) must be greater than or equal to \(6\), the domain of the function can be expressed in interval notation as:\[ [6, \infty) \].

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Key Concepts

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

square root function
The square root function is crucial in mathematics, mainly when determining the domain of a function. The square root of a number is a value that, when multiplied by itself, gives the original number.
For instance, the square root of 9 is 3, because 3 multiplied by 3 is 9. In mathematical notation, the square root of a number 'a' is written as \( \sqrt{a} \).
However, the square root is only defined for non-negative numbers. This means we cannot take the square root of a negative number in the real number system.
  • For example, \( \sqrt{-4} \) is not a real number.
  • Thus, whenever you encounter a square root, check if the number inside (the radicand) is non-negative.
In the case of the function \( f(x)=\sqrt{x-6} \), this means that \( x-6 \) must be zero or positive.
inequality solving
Inequality solving is a vital technique to determine the domain of functions involving square roots. An inequality represents a relationship where one value is larger or smaller than another. In this context, it ensures that the expression inside the square root is non-negative.
To solve inequalities like \( x-6 \geq 0\), follow these steps:
  • Isolate the variable (in this case, x) on one side of the inequality.
  • For \( x-6 \geq 0\), add 6 to both sides to get \( x \geq 6\).
This inequality tells you the smallest value x can take, ensuring the function remains defined. By solving the inequality, you determine that x must be at least 6. Therefore, any value of x less than 6 will not satisfy the function \( f(x) = \sqrt{x-6} \).
interval notation
Interval notation is an efficient way to express sets of numbers, often used to describe the domain of a function. It uses brackets and parentheses to denote the start and end of an interval.
When you solved the inequality \( x \geq 6 \), you found that x can be any number greater than or equal to 6. In interval notation, we express this as \( [6, \infty) \).
Here鈥檚 a quick guide:
  • Square brackets [ ] denote that the endpoint is included in the interval (closed interval).
  • Parentheses ( ) indicate that the endpoint is not included (open interval).
  • The interval \( [6, \infty) \) means that the domain starts at 6 and extends to infinity, including 6 but not infinity (as infinity is not a number).
Using interval notation simplifies communicating the range of possible values for x in functions, offering clear and concise domain representation.

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