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Chapter 12: Problem 85

Determine the domain of each function. $$p(a)=\sqrt{a-8}$$

Short Answer

Expert verified
The domain of the function \(p(a) = \sqrt{a-8}\) is given by the inequality \(a \ge 8\), which in set notation can be written as: \(D(p) = \{a | a \ge 8\}\).

Step by step solution

01

Write down the inequality representing the domain of the function

The square root function is only defined for non-negative values. Therefore, to find the domain of the function, we need to solve the inequality \(a-8 \ge 0\).
02

Solve the inequality for \(a\)

To solve the inequality \(a-8 \ge 0\), we need to move the constant term to the other side of the inequality to isolate \(a\). Add \(8\) to both sides of the inequality: \(a \ge 8\)
03

Write the domain

The inequality \(a \ge 8\) represents the domain of the function. Using set notation, the domain can be written as: \[D(p) = \{a | a \ge 8\}\]

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Key Concepts

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

Square Root Function
A square root function, like the one we have in the exercise, takes the form \( \sqrt{x} \). It produces the principal (non-negative) square root of a given number. The domain of a square root function includes all values for which the expression under the square root is non-negative. This is because the square root of a negative number is not defined within the set of real numbers. In our given problem, the function is \( p(a) = \sqrt{a-8} \), so we need \( a-8 \geq 0 \) to ensure that what's inside the square root remains non-negative. This constraint shapes the possible inputs (domain) of the function.
Inequalities
In the context of finding the domain of the function, inequalities can be used to express the conditions necessary for the function to have real number outputs. An inequality is a mathematical statement that relates expressions that are not exactly equal but instead greater or less than each other. With the inequality \( a-8 \geq 0 \), we're saying that \( a \) must be at least 8 for the function \( p(a) \) to produce a real number. Solving this involves isolating \( a \), which implies adding 8 to both sides, thus giving us \( a \geq 8 \). This solution tells us about the possible values of \( a \) that make the function work without producing errors.
Set Notation
Set notation provides a clear and concise way to express the domain of a function. The notation \( \{a \mid a \ge 8\} \) describes the set of all numbers \( a \) that satisfy the condition \( a \geq 8 \). In words, this means "the set of all \( a \) such that \( a \) is greater than or equal to 8." Set notation is very useful in mathematics as it can effectively summarize a range of values or conditions that apply to an operation or function. In this exercise, we've used set notation to define the domain of the square root function clearly and unambiguously.

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Most popular questions from this chapter

Graph the following piecewise functions. $$g(x)=\left\\{\begin{aligned}x-1, & x \geq 2 \\\\-3 x+3, & x<2\end{aligned}\right.$$

If the following transformations are performed on the graph of \(f(x)\) to obtain the graph of \(g(x),\) write the equation of \(g(x)\). \(f(x)=x^{2}\) is shifted left 3 units and up \(\frac{1}{2}\) unit.

For each pair of functions, find a) \(\left(\frac{f}{g}\right)(x)\) and b \(\left(\frac{f}{g}\right)(-2)\) Identify any values that are not in the domain of \(\left(\frac{f}{g}\right)(x)\). $$f(x)=x^{2}-5 x-24, g(x)=x-8$$

Graph the following piecewise functions. $$k(x)=\left\\{\begin{array}{ll}\frac{1}{2} x+\frac{5}{2}, & x<3 \\\\-x+7, & x \geq 3\end{array}\right.$$

For each pair of functions, find a) \(\left(\frac{f}{g}\right)(x)\) and b \(\left(\frac{f}{g}\right)(-2)\) Identify any values that are not in the domain of \(\left(\frac{f}{g}\right)(x)\). $$f(x)=3 x^{2}+14 x+8, g(x)=3 x+2$$
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