Problem 91 - Beginning and Intermediate Algebra

Question

Determine the domain of each function. $$r(a)=\sqrt{9-a}$$

Step-by-Step Solution

Answer

The domain of the function \(r(a)=\sqrt{9-a}\) is \(a \in (-\infty, 9]\).

Step 1: Set up inequality

To find the domain for the function, we need to find all the values for which the expression inside the square root is non-negative. So, we start by setting up the inequality: $$9 - a \ge 0$$

Step 2: Solve the inequality

Next, solve the inequality (add 'a' to both sides of the inequality): $$9 \ge a$$

Step 3: Express the domain

Now that we have found the inequality for the function, we can express the domain of the function in interval notation. Since 'a' is less than or equal to 9, the domain is: $$a \in (-\infty, 9]$$

Key Concepts

Inequality Solving

When dealing with square root functions, it is vital to solve inequalities in order to determine the domain of the function. The domain refers to the set of all possible input values for which the function is defined. For square root functions, the expression inside the root, known as the radicand, must be non-negative. This means

• The radicand should be greater than or equal to zero.
• Set up an inequality based on the condition that the radicand is non-negative.
• For our function, \( r(a) = \sqrt{9-a} \), we set the inequality \( 9-a \geq 0 \).

By setting this inequality, we ensure that the square root will have "real" number values, avoiding the undefined scenarios that occur with negative numbers within a square root.

Square Root Functions

Square root functions are special types of functions defined by a square root expression. They have specific characteristics, and one of the most important ones is that they are only defined for non-negative radicands. This is because the square root of a negative number is not a real number, making it crucial to identify when they're valid.

• The general form is \( f(x) = \sqrt{x} \).
• In the case of \( r(a) = \sqrt{9-a} \), the function is only defined when \(9-a \geq 0\).
• Square root functions will graph as a curve starting at the point where the inside of the root equals zero.
• The domain of a square root function is linked directly to solving the inequality of the radicand.

These points help ensure that the function results in real, meaningful outputs.

Interval Notation

Interval notation is a way of representing a set of values, typically used to indicate domain or range for functions. It provides a compact way to describe the range of inputs or outputs in a continuous setting.Here's how you can understand it better:

• It uses brackets \([]\) and parentheses \(()\) to show closed and open intervals respectively.
• A closed interval \([a, b]\) means the boundary values \(a\) and \(b\) are included, whereas \((a, b)\) means they are not included.
• For the function \( r(a) = \sqrt{9-a} \), solving \(9-a \geq 0\) gives \(a \leq 9\).
• Thus, the domain is expressed as \(a \in (-\infty, 9]\), capturing all potential values 'a' can take.

Using interval notation offers a neat and precise way to convey domains and ranges, especially when they include infinite sets of numbers.