Problem 11 - Algebra and Trigonometry

Question

An expression is given. (a) Evaluate it at the given value. (b) Find its domain. $$ \sqrt{x+3}, \quad x=6 $$

Step-by-Step Solution

Answer

The expression evaluates to 3 when \( x=6 \). The domain is \( x \geq -3 \).

Step 1: Substitute the Value into the Expression

We are given the expression \( \sqrt{x+3} \) and need to evaluate it for \( x = 6 \). We substitute 6 for \( x \) in the expression: \( \sqrt{6+3} \).

Step 2: Simplify the Expression

Simplify the expression \( \sqrt{6+3} \) by adding the numbers inside the square root: \( \sqrt{9} \).

Step 3: Evaluate the Square Root

Find the square root of 9. We know that \( \sqrt{9} = 3 \). So, the evaluated expression at \( x = 6 \) is 3.

Step 4: Determine the Domain of the Expression

The expression is \( \sqrt{x+3} \) which is defined only when the expression inside the square root is non-negative. Therefore, we need \( x+3 \geq 0 \).

Step 5: Solve the Inequality for the Domain

Solve the inequality \( x + 3 \geq 0 \) by subtracting 3 from both sides to get \( x \geq -3 \). This means the domain of \( \sqrt{x+3} \) is \( x \geq -3 \).

Key Concepts

Square Root Function

A square root function is a type of mathematical function where the output is the square root of the input expression. Square root functions often look like this: \( \sqrt{x} \). In the example expression \( \sqrt{x+3} \), the operation is performed on \( x+3 \), meaning you first add 3 to the value of \( x \) and then take the square root of the result.

The square root function can only operate on values that are non-negative. This is because the square root of a negative number isn't defined within the real number system. So, whenever you encounter a square root function, it's important to ensure your input value or expression (like \( x+3 \)) is 0 or greater. This non-negative requirement helps determine the domain of square root functions, as they'll only be defined for inputs that meet this criteria.

Evaluating Expressions

Evaluating an expression means calculating its value when the variable is replaced by a specific number. To evaluate the expression \( \sqrt{x+3} \) when \( x=6 \), we substitute 6 into the place of \( x \) in the expression.

Here are the steps:

• Substitute the given value: Insert 6 for \( x \), resulting in \( \sqrt{6+3} \).
• Simplify inside the square root: Add 6 and 3, which equals 9, so you have \( \sqrt{9} \).
• Compute the square root: The square root of 9 is 3.

This sequence of steps clearly shows the value of the expression when the specific value is used. Evaluating expressions is critical in algebra as it allows us to see how an expression behaves for different values of the variable.

Inequality Solutions

In the context of finding a domain, solving inequalities helps you figure out the set of values for which an expression is defined. For the square root function \( \sqrt{x+3} \), you must ensure the inside of the square root, \( x+3 \), is non-negative. Meaning, \( x+3 \geq 0 \).

To solve this inequality:

• Start with the inequality: \( x+3 \geq 0 \).
• Subtract 3 from both sides: This gives \( x \geq -3 \).

The solution, \( x \geq -3 \), indicates the domain of the function. This specific inequality solution tells you for which values of \( x \) the square root function will yield a real number.

Understanding and solving inequalities is fundamental when dealing with domains, ensuring that you only work within the range where the function is properly defined.