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Chapter 2: Problem 21

In Exercises 1鈥�30, find the domain of each function. $$ g(x)=\sqrt{5 x+35} $$

Short Answer

Expert verified
The domain of the function \(g(x)=\sqrt{5x+35}\) is \(x \geq -7\).

Step by step solution

01

Understanding the Function

Consider the function \( g(x)=\sqrt{5x+35} \). The square root function requires a non-negative input, as square roots of negative numbers are not defined for real numbers. Therefore, the term inside the square root, \(5x + 35\), must be greater than or equal to zero.
02

Setting Up the Inequality

The term inside the square root, \(5x + 35\), must be greater than or equal to zero. We set this up as an inequality: \(5x + 35 \geq 0\).
03

Solve the Inequality

To isolate \(x\), subtract 35 from both sides of the inequality, resulting in \(5x \geq -35\). Then, divide both sides by 5, yielding \(x \geq -7\).

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Key Concepts

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

Inequality Solving
Inequality solving is an essential skill when determining the domain of functions that include conditions like square roots. An inequality shows the relationship between two expressions and indicates how one value compares to another. For solving an inequality, we perform similar steps as we do for solving equations, but with a focus on maintaining the inequality direction.
Here is how you can solve inequalities:
  • Identify any operations needed to isolate the variable on one side.
  • Apply inverse operations like addition or subtraction and multiplication or division as you would in a regular equation.
  • Be mindful that multiplying or dividing both sides by a negative number reverses the inequality sign.
Inequalities allow us to express sets of values, often forming intervals. Solving them plays a crucial role, especially when dealing with functions like square roots.
Square Root Function
The square root function is fundamental when dealing with non-negative domains in mathematics. In a square root function like \(g(x) = \sqrt{5x + 35}\), the expression under the square root (called the radicand) must be non-negative to ensure the output is a real number.
Why does this matter?
  • Square roots of negative numbers don't exist in the set of real numbers because no real number multiplied by itself gives a negative result.
  • This restriction means that for real-valued functions, the radicand must be zero or positive, establishing the relationship to inequality solving.
Thus, when finding the domain of a square root function, one always begins by setting up an inequality for the radicand to be non-negative.
Real Numbers Domain
The domain of a function refers to all the input values (x-values) for which the function is defined. In the context of real numbers, this often means ensuring that all operations within the function return real numbers under given conditions.
Understanding the domain:
  • For the square root function \(g(x) = \sqrt{5x + 35}\), calculations in the square root must not lead to undefined terms, which means working within real numbers only.
  • Real numbers include all numbers on the continuous number line, accepting integers, fractions, and irrational numbers, but not imaginary numbers.
  • By solving the inequality \(5x + 35 \geq 0\), we determined that \(x \geq -7\) ensures that the radicand remains non-negative, hence within the real numbers.
Therefore, focusing on the real numbers' domain is crucial in defining acceptable input values for the function and ensuring that the function's operation remains valid.

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Most popular questions from this chapter

Exercises \(103-105\) will help you prepare for the material covered in the next section. Let \(\left(x_{1}, y_{1}\right)=(7,2) \quad\) and \(\quad\left(x_{2}, y_{2}\right)=(1,-1) . \quad\) Find \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}} .\) Express the answer in simplified radical form.

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

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. A tangent line to a circle is a line that intersects the circle at exactly one point. The tangent line is perpendicular to the radius of the circle at this point of contact. Write an equation in point-slope form for the line tangent to the circle whose equation is \(x^{2}+y^{2}=25\) at the point \((3,-4)\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Find the area of the donut-shaped region bounded by the graphs of \((x-2)^{2}+(y+3)^{2}=25\) and \((x-2)^{2}+(y+3)^{2}=36\)

Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$x^{2}+y^{2}-4 x-12 y-9=0$$
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