Problem 92 - Beginning and Intermediate Algebra

Question

Determine the domain of each function. $$g(c)=\sqrt{8-5 c}$$

Step-by-Step Solution

Answer

The domain of the function \(g(c) = \sqrt{8 - 5c}\) is \(c \in (-\infty, \frac{8}{5}]\).

Step 1: Identify expression under the square root

In this exercise, we need to analyze the expression under the square root: \(8 - 5c\). We want to find out for which values of \(c\) the expression is greater than or equal to zero.

Step 2: Form an inequality

We want the expression under the square root to be non-negative, so we'll set up an inequality involving the expression: \(8 - 5c \geq 0\).

Step 3: Solve the inequality

Now, let's solve the inequality for the variable \(c\): \begin{align} 8 - 5c &\geq 0\\ 8 &\geq 5c\\ \frac{8}{5} &\geq c \end{align}

Step 4: Write the domain

Based on the inequality we found, \(c\) must be less than or equal to \(\frac{8}{5}\). In interval notation, this is represented as: \[c \in (-\infty, \frac{8}{5}]\] Thus, the domain of the function \(g(c) = \sqrt{8 - 5c}\) is \(c \in (-\infty, \frac{8}{5}]\).

Key Concepts

Square Root Function

A square root function is a type of mathematical function that contains a square root. It is usually expressed in the form \(g(c) = \sqrt{8-5c}\). The square root function only returns non-negative results since the square root of a negative number is not defined in the real numbers. This property directly affects the domain of the square root function that we need to determine.

The domain of a function is simply the set of input values for which the function is defined. For square root functions, this means finding values where the expression under the square root is greater than or equal to zero, since negative values would make the function undefined in the set of real numbers.

Inequality Solving

Inequality solving involves finding the range of values that satisfy an inequality. In the case of square root functions, we deal with the inequality that ensures the expression under the square root remains non-negative.

Taking the function \(g(c) = \sqrt{8-5c}\), we start by setting up the inequality:

• \(8 - 5c \geq 0\)

Next, we solve this inequality step by step to find allowable values for \(c\):

• First, manipulate the inequality to isolate \(c\): \(8 \geq 5c\).
• Then, divide by 5 to get: \(\frac{8}{5} \geq c\).

This tells us that \(c\) should be less than or equal to \(\frac{8}{5}\) for the inequality to hold.

Interval Notation

Interval notation is a way of writing subsets of the real numbers, illustrating the domain or range of a function in a simple, compact manner. For the function \(g(c) = \sqrt{8-5c}\), we determined through inequality solving that \(c\) must satisfy \(c \leq \frac{8}{5}\).

With interval notation, we express this as:

• \[c \in (-\infty, \frac{8}{5}]\]

The notation \((-\infty, \frac{8}{5}]\) represents all real numbers less than or equal to \(\frac{8}{5}\). The parentheses \((\) indicate that \(-\infty\) is not included, while the bracket \([]\) indicates that \(\frac{8}{5}\) is included in the domain.

Real Numbers

Real numbers encompass all the numbers that can be found on the number line, including both rational and irrational numbers. They are an extensive set of numbers and are usually denoted by the symbol \(\mathbb{R}\). The domain of the square root function \(g(c) = \sqrt{8-5c}\) evaluated above falls within the set of real numbers.

However, because the expression inside the square root should be non-negative, not every real number is part of the domain. Only those real numbers that satisfy the condition \(c \leq \frac{8}{5}\) are included. This concept is important because real numbers are the building block for a lot of number sets and understanding the restrictions or boundaries each function imposes on variables is key to solving mathematical problems.