Problem 111 - Algebra and Trigonometry

Question

Find the domain of \(h(x)=\sqrt{36-2 x}\)

Step-by-Step Solution

Answer

The domain of the function \(h(x)=\sqrt{36-2x}\) is \(x \leq 18\).

Step 1: Identify the condition

The key condition here is that the term under the square root, that is \(36-2x\), must be greater than or equal to zero, as you cannot take the square root of a negative number. So set up the following inequality: \(36 - 2x \geq 0\).

Step 2: Solve the inequality

Now solve the inequality \(36 - 2x \geq 0\) for \(x\). First, move \(2x\) to the other side of the inequality: \(36 \geq 2x\). Then, divide by 2 to solve for \(x\): \(x \leq 18\).

Step 3: Write down the domain

The solution of the inequality is, in the notational form, \(x \leq 18\), which means that \(x\) can be any number up to and including 18. This is the domain of the function \(h(x)=\sqrt{36-2x}\).

Key Concepts

Square Root Function

A square root function is one type of function involving the mathematical operation of taking a square root. When you see a function like \( h(x) = \sqrt{36 - 2x} \), we consider this a square root function because it includes a square root operation. What's important about square roots is that you can only take the square root of non-negative numbers. Hence, in such functions, the expression inside the square root (called the radicand) must be zero or positive.

For the function \( h(x) = \sqrt{36 - 2x} \), it means the radicand \( 36 - 2x \) must satisfy the condition \( 36 - 2x \geq 0 \). This is crucial because square roots of negative numbers are not defined in the set of real numbers, which we typically work with when determining a function's domain.

• **Radicand**: The expression inside the square root
• **Function Domain**: The set of input values for which the function is defined

In short, understanding the nature of the square root function helps us define where the function exists, i.e., find its domain based on algebraic conditions.

Inequalities in Algebra

Inequalities in algebra are expressions that describe a relationship of greater than or less than between two expressions. Unlike equations that assert equality, inequalities show when one side is greater or smaller. They are often represented using symbols like \( >, <, \geq, \leq \).

In the context of our function \( h(x) = \sqrt{36 - 2x} \), we deal with the inequality \( 36 - 2x \geq 0 \). This inequality results from imposing the condition that the radicand must not be negative to ensure the square root function is valid.

• **Solving Inequalities**: Similar to solving equations, manipulate both sides to find the range of values.
• **Reversing Inequality Signs**: An important rule when multiplying or dividing by a negative number is that you must flip the inequality sign.

Inequalities help in determining what values \( x \) can take to make expressions valid, which directly helps us find the domain of functions involving square roots.

Solving Inequalities

Solving inequalities involves finding the set of values for the variable that makes the inequality true. There are systematic steps similar to solving equations but with particular rules related to inequalities.

To solve \( 36 - 2x \geq 0 \), we first rearrange to isolate terms with \( x \):

• Add or subtract values as needed: \( 36 \geq 2x\)
• Divide both sides by 2: \( x \leq 18\)

With inequalities, it’s crucial to remember:

• **Direction of Inequality**: When dividing or multiplying by a negative number, reverse the inequality.
• **Checking Solutions**: Always verify that values satisfy the original inequality.

In concluding that \( x \leq 18 \), we define the domain as every \( x \) up to and including 18. By understanding these steps, you can solve any simple linear inequality, helping determine valid inputs for a variety of functions.