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Chapter 3: Problem 72

State the domain of \(g(x)=\sqrt{x-2} .[1.3]\)

Short Answer

Expert verified
The domain of the function \(g(x) = \sqrt{x-2}\) is \(x \geq 2\). All real numbers equal to or greater than 2 are the possible values for x that the function g(x) can accept.

Step by step solution

01

Identify the square root function

In this exercise, the function given is \[g(x) = \sqrt{x-2}\]. The square root function is identified by the square root sign.
02

Set the radicand greater than or equal to zero

The radicand in the function \[g(x) = \sqrt{x-2}\] is \((x - 2)\). For the square root function to have real values, the radicand should be greater than or equal to zero. Thus, set up the inequality \((x - 2) \geq 0\].
03

Solve for x

Solve the inequality \((x - 2) \geq 0\) for x. Adding 2 to both sides gives \(x \geq 2\).

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Key Concepts

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

Understanding the Square Root Function
When we talk about a square root function, we are dealing with a special type of mathematical operation that finds a value, which when multiplied by itself, gives the original number under the square root sign. In the context of the function \(g(x) = \sqrt{x-2}\), the square root is applied to the expression \(x-2\).

For the function to be valid for all real numbers, the value inside the square root, known as the radicand, must be non-negative because the square root of a negative number is not a real number (it's an imaginary number). This is why, when determining the domain of a square root function, we ensure that the radicand is greater than or equal to zero. This guarantees that the function's values are real numbers, which is a fundamental requirement in real-world applications.
Radicand Inequality for Domain Determination
The concept of radicand inequality is closely related to the domain of square root functions. The radicand is the number or expression inside the square root, and its value dictates whether a square root will produce a real number as its output. In the function \(g(x) = \sqrt{x-2}\), the radicand is \(x-2\).

To find the domain, we need to identify for which x-values the radicand will be zero or positive. This is because the square root of a negative number is not defined within the real number system. We thus establish the inequality \(x - 2 \geq 0\), ensuring that the radicand will not be negative for any value within the domain. This step is crucial for determining the set of all possible inputs (x-values) that the function can accept and still produce a real number output.
Solving Inequalities to Find Function Domains
Solving inequalities is a critical skill when determining the domain of a function, especially when the function includes a square root. Unlike a straightforward equation, solving an inequality does not provide a single solution but instead a range of values that satisfy the inequality's conditions.

In the example of \(g(x) = \sqrt{x-2}\), the inequality \(x - 2 \geq 0\) tells us that the domain of the function includes all x-values greater than or equal to 2. To solve this inequality, you simply isolate the variable x on one side, which in our case leads to \(x \geq 2\). Thus, the domain of the square root function is the set of all real numbers greater than or equal to 2. This informs us that the function \(g(x)\) only takes on real values for these x-values, ensuring the outputs are grounded in the real number system.

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Most popular questions from this chapter

Crude oil leaks from a tank at a rate that depends on the amount of oil that remains in the tank. Because \(\frac{1}{8}\) of the oil in the tank leaks out every 2 hours, the volume of oil \(V(t)\) in the tank after \(t\) hours is given by \(V(t)=V_{0}(0.875)^{1 / 2},\) where \(V_{0}=350,000\) gallons is the number of gallons in the tank at the time the tank started to leak \((t=0)\) a. How many gallons does the tank hold after 3 hours? b. How many gallons does the tank hold after 5 hours? c. How long, to the nearest hour, will it take until \(90 \%\) of the oil has leaked from the tank?

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Explain how to use the graph of the first function \(f\) to produce the graph of the second function \(F\). $$f(x)=10^{x}, F(x)=10^{x-2}$$
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