Problem 84 - Beginning and Intermediate Algebra

Question

Determine the domain of each function. $$g(c)=\sqrt{c+10}$$

Step-by-Step Solution

Answer

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

Step 1: Identify the constraint on the input value 'c'

We are dealing with a square root function, and we know the argument inside the square root should be non-negative to be a real number. The argument here is \(c+10\). So, we need to find the values of c for which \(c+10 \geq 0\).

Step 2: Solve the inequality to find the domain of the function

To find the domain, solve the inequality \(c+10 \geq 0\). Subtract 10 from both sides of the inequality: \(c \geq -10\) So the domain of the function is \(c \geq -10\), which can be written in interval notation as \([-10,\infty)\). The domain of the function \(g(c) = \sqrt{c+10}\) is \([-10,\infty)\).

Key Concepts

Square Root Function

Square root functions are a fascinating type of function in mathematics. They are defined by the square root of an expression. Here, the function is represented as \( g(c) = \sqrt{c+10} \). For square root functions, it is essential that the value inside the square root, known as the radicand, stays non-negative. This ensures the function outputs real numbers. Thus, everything underneath the square root symbol must be zero or positive.

• Radicand: The expression inside the square root, which is \( c+10 \) in this case.
• Non-negative Requirement: Set the radicand \( c+10 \geq 0 \).

Ensuring the radicand is non-negative helps us determine valid input values for the function. Embracing this non-negativity condition is foundational when working with square root functions.

Inequalities

Inequalities are mathematical expressions involving comparison. They show relationships like greater than, less than, equal to, and others. Here, our focus is on the inequality \( c+10 \geq 0 \). This expression tells us which values of \( c \) are acceptable.
Inequalities use symbols such as:

• \( \geq \) : greater than or equal to
• \( \leq \) : less than or equal to

To solve the inequality \( c+10 \geq 0 \), we perform operations just like equations. Subtraction maintains the inequality direction, so subtract 10 from both sides:\[c \geq -10\]This solution means that \( c \) must be greater than or equal to \(-10\). Solving inequalities provides a range of values for which the function is defined.

Interval Notation

Interval notation is a concise way to indicate a set of values that satisfy an inequality. For our function \( g(c) = \sqrt{c+10} \), the range of acceptable values for \( c \) begins from \(-10\) and continues to infinity. Interval notation helps present these values compactly.
Here's how to interpret interval notation:

• \([-10, \infty)\): All values from \(-10\) inclusive, to infinity. The bracket \([\) means \(-10\) is included.
• Parentheses \((\) indicate exclusion. For infinity, we use a parenthesis because infinity is not a specific number.

Using interval notation, we express intervals clearly and efficiently, summing up complex inequalities into a simple form.