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Chapter 2: Problem 19

In Exercises 1鈥�30, find the domain of each function. $$ g(x)=\frac{1}{\sqrt{x-3}} $$

Short Answer

Expert verified
The domain of the function g(x) = \( \frac{1}{\sqrt{x-3}} \) is all real numbers greater than 3. In interval notation, this is written as \( (3, \infty) \).

Step by step solution

01

Express the domain

First we need to express the condition for the domain of function g(x). We know that the denominator cannot be zero or negative. Therefore, the condition for the domain is: \(x-3 > 0\)
02

Solve the inequality

To find the values that x can take, we need to solve the inequality obtained in the previous step. In order to do so, simply add 3 on both sides of the inequality to isolate x: \(x > 3\)
03

Express the domain of g(x)

The solution to the inequality tells us the domain of the function g(x). Therefore, any real number greater than 3 is in the domain of g(x). In interval notation, this is expressed as: \( (3, \infty) \)

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Key Concepts

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

Understanding Inequality Solving
Inequalities are mathematical expressions that show the relationship between two values using signs like \(<\), \(>\), \(\leq\), and \(\geq\). Solving inequalities involves finding the set of values that satisfy the condition stated by the inequality.
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The given expression \(x - 3 > 0\) is an inequality. Our task is to find the values of \(x\) that make this expression true. Here鈥檚 how it works:
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  • Add 3 to both sides of the inequality to isolate \(x\).
  • This results in \(x > 3\), meaning \(x\) must be greater than 3.
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By following these steps, we've determined the set of possible \(x\) values, which is crucial for understanding the domain of the function.
Exploring Radical Expressions
Radical expressions involve roots, like square roots or cube roots. The function \(g(x)=\frac{1}{\sqrt{x-3}}\) includes a square root in its denominator.
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Understanding radical expressions is essential because they affect the domain of a function.
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Here's what we need to consider:
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  • The expression under the square root, \(x-3\), must be positive. Square roots of negative numbers aren鈥檛 real.
  • This means: \(x-3 > 0\).
\
By ensuring \(x-3\) is positive, we're making sure the square root is defined with real numbers. This directly influences the values \(x\) can take, shaping the function's domain.
Demystifying Function Notation
Function notation, like \(g(x)\), is a way to represent functions in a structured form. It tells us the function's relationship with the variable \(x\).
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The function \(g(x)\) shows that for each value of \(x\), there's a corresponding value of \(g\). Here鈥檚 what to keep in mind:
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  • The notation \(g(x)\) specifies the input \(x\) and the resulting output \(g\).
  • This notation is useful for expressing the function's rule or equation; in this case, \(g(x)=\frac{1}{\sqrt{x-3}}\).
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Using function notation allows us to clearly communicate and work with the function, making it easier to discuss the domain and other properties.
Navigating Interval Notation
Interval notation is a concise way to describe the set of numbers that form the domain of a function. Instead of writing out all possible values, we use symbols to indicate ranges.
\
For the function \(g(x)\), we determined that \(x > 3\). This is expressed in interval notation as \((3, \infty)\). Here's how interval notation works:
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  • The parenthesis \(()\) indicates that 3 is not included in the domain.
  • \(\infty\) represents infinity, showing that \(x\) can be any number greater than 3.
\
This approach succinctly summarizes the domain, helping us quickly identify the valid inputs for the function.

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Most popular questions from this chapter

Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$x^{2}+y^{2}+3 x-2 y-1=0$$

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