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Chapter 3: Problem 101

Write the domain of the function in interval notation. $$ h(x)=\sqrt{\frac{3 x}{x+2}} $$

Short Answer

Expert verified
[-2,鈭�) except x = 0

Step by step solution

01

- Identify the domain restrictions

The domain of the function is determined by the values of x for which the expression inside the square root is non-negative and the denominator is not zero.
02

- Set the radicand greater than or equal to zero

Solve the inequality for the radicand to be non-negative:
03

- Solve the inequality

First, note that the expression inside the square root is $

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Key Concepts

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

Interval Notation
Interval notation is a way of expressing a set of numbers (interval) on the real number line. It uses brackets:
  • Square brackets [ ] include the endpoints of the interval.
  • Round brackets ( ) exclude the endpoints.
For example, the interval [1, 5) includes all numbers from 1 to 5, including 1 but excluding 5. It's a concise way to represent domains of functions, solutions to inequalities, and more. In the context of our function, we need interval notation to state where the function is defined. This means figuring out all x-values that make the function valid.
Radicand
In a square root function, the expression inside the square root symbol is called the radicand. For our function, the radicand is the fraction \( \frac{3x}{x+2} \). The key rule for radicands in square roots is that they must be non-negative. This means: \(\frac{3x}{x+2} \geq 0 \). Solving this inequality helps determine the values of \(x\) for which the function is defined. We also need to ensure the denominator is not zero, since division by zero is undefined.
Inequalities
Inequalities are mathematical statements that compare two values or expressions. They are critical when finding the domain of functions involving square roots and fractions. For \( h(x) = \sqrt{\frac{3x}{x+2}} \), we need to solve the inequality \( \frac{3x}{x+2} \geq 0 \). This involves determining when the numerator and denominator are either both positive or both negative. After solving these inequalities, we use interval notation to summarize the valid \(x\) values. Ensuring a solid understanding of inequalities is essential for mastering these types of problems.

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Most popular questions from this chapter

The yearly membership for a professional organization is \(\$ 250\) per year for the current year and increases by \(\$ 25\) per year. If a person joins for \(x\) consecutive years, the average cost per year \(\overline{C_{1}}(x)\) (in \$) is given by $$\overline{C_{1}}(x)=\frac{475+25 x}{2}$$ a. Find the average cost per year if a person joins for \(5 \mathrm{yr}, 10 \mathrm{yr},\) and \(15 \mathrm{yr}\) b. The professional organization also offers a one-time fee of \(\$ 2000\) for a lifetime membership. If a person purchases a lifetime membership, write an average cost function representing the average cost per year \(\overline{C_{2}}(x)\) (in \$) for \(x\) years of membership. c. If a person purchases a lifetime membership, compute the average cost per year for \(5 \mathrm{yr}, 10 \mathrm{yr},\) and \(15 \mathrm{yr}\). d. Interpret the meaning of the horizontal asymptote for the graph of \(y=\overline{C_{2}}(x)\)

Write the domain of the function in interval notation. $$ p(x)=\sqrt{2 x^{2}+9 x-18} $$

A power company burns coal to generate electricity. The cost \(C(x)\) (in \(\$ 1000\) ) to remove \(x \%\) of the air pollutants is given by $$C(x)=\frac{600 x}{100-x}$$ a. Compute the cost to remove \(25 \%\) of the air pollutants. $$(\text { Hint }: x=25 .)$$ b. Determine the cost to remove \(50 \%, 75 \%,\) and \(90 \%\) of the air pollutants. c. If the power company budgets \(\$ 1.4\) million for pollution control, what percentage of the air pollutants can be removed?

Determine if the statement is true or false. Show that \(x-a\) is a factor of \(x^{n}-a^{n}\) for any positive integer \(n\) and constant \(a\).

Suppose that \(y\) varies directly as \(x^{5}\) and inversely as \(w^{2}\). If both \(x\) and \(w\) are doubled, what is the effect on \(y ?\)
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