Problem 68 - College Algebra

Question

Find the domain of each function. $$ g(x)=\sqrt{7 x-70} $$

Step-by-Step Solution

Answer

The domain of the function \( g(x) = \sqrt{7x - 70} \) is \([10, \infty)\).

Step 1: Set the inside of the square root to be greater than or equal to zero

Set up the inequality \(7x - 70 \geq 0\)

Step 2: Solve the inequality for x

Add 70 to both sides to get \(7x \geq 70\). Then divide by 7 to get \(x \geq 10\).

Step 3: Write the domain

The domain of the function is all x such that \(x \geq 10\), or in interval notation, \([10, \infty)\).

Key Concepts

Understanding Inequality Solving

Inequality solving is a fundamental skill in mathematics, often used when dealing with functions, especially when determining their domain. Let’s walk through the process of solving an inequality step-by-step.

Consider the inequality given in the expression for the domain of the square root function:

• First, you set up the inequality from the inside of the square root: \(7x - 70 \geq 0\).
• Add 70 to both sides to balance the inequality: \(7x \geq 70\).
• Finally, divide by 7 to solve for \(x\), yielding \(x \geq 10\).

This inequality tells us the values of \(x\) that keep the expression inside the square root greater than or equal to zero, ensuring we have real numbers under the square root.

Understanding how to solve inequalities effectively is crucial when finding the domain of functions, as it allows us to determine which inputs make the function valid.

Exploring the Square Root Function

The square root function is a common type of function that extracts the square root of a given expression or number. To ensure the function produces real numbers, the expression inside the square root must be non-negative.

Why is this important?

Because square roots of negative numbers result in imaginary numbers rather than real numbers. For the function \(g(x)=\sqrt{7x-70}\), this means:

• The expression \(7x - 70\) must be greater than or equal to zero.
• This restriction stems from the properties of square roots.

The square root function is pivotal in many mathematical and real-world applications because it is used to find values associated with areas, magnitudes, and more. Understanding their properties and restrictions helps in solving complex mathematical problems efficiently.

Utilizing Interval Notation

Interval notation is a concise way to represent the domain or range of a function. It uses brackets and parenthesis to indicate the start and end points of intervals on a number line.

In our example, once the inequality \(x \geq 10\) is solved, we express this solution using interval notation:

• A square bracket \([\) indicates that 10 is included in the domain \([10,\).
• An infinity symbol \(()\) indicates that the domain extends indefinitely without bound \([10, \infty)\).

Interval notation is crucial as it provides an easy-to-understand visual cue for the range or domain of functions. This method is widely used in mathematics because it neatly and clearly conveys which values are acceptable for a variable, especially for continuous domains like those associated with inequalities. Mastering interval notation will be immensely helpful not only in solving textbook problems but also in analyzing real-world situations involving continuous data.