Problem 5 - Algebra and Trigonometry

Question

\(1-6=\) 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

3; Domain is all \( x \geq -3 \).

Step 1: Substitute the Value

Substitute the given value of \(x = 6\) into the expression \(\sqrt{x + 3}\). This will give us \(\sqrt{6 + 3}\).

Step 2: Simplify the Expression

Perform the arithmetic operation inside the square root. \(6 + 3 = 9\). Then take the square root, \(\sqrt{9} = 3\).

Step 3: Determine Expression's Domain

For the expression \(\sqrt{x + 3}\) to be valid, the expression under the square root must be non-negative. Hence, \(x + 3 \geq 0\). Solve the inequality: \(x \geq -3\).

Key Concepts

Square Root Function

Square root functions involve expressions that are under the radical sign, symbolized as \( \sqrt{} \). These functions deal with understanding how to find the value or the possible range of values (domain) that yield real numbers when the square root is calculated. For example, in the expression \( \sqrt{x+3} \), the part \( x+3 \) is under the square root. The square root function results in a real number only if \( x+3 \) is non-negative, that is, \( x+3 \geq 0 \). This is important because you cannot take the square root of a negative number in the set of real numbers without involving imaginary numbers.
To evaluate a square root function, substitute the given value for \( x \) into the expression and solve for the square root. Simplifying \( \sqrt{6+3} \) results in \( \sqrt{9} \), which is equal to 3. The radical sign represents the principal square root, meaning the non-negative root.

Inequalities

Inequalities are mathematical statements used to compare the relative sizes or values of two expressions. They indicate whether one expression is less than, greater than, or equal to another expression. In the context of square root functions, inequalities help determine the domain of the function. The domain is the set of all possible inputs (or \( x \) values) that make the function output real numbers.
For a function like \( \sqrt{x+3} \), you need to solve the inequality \( x+3 \geq 0 \) to find the domain. This is because square roots require non-negative radicands in real number calculations. Solving \( x+3 \geq 0 \) gives us \( x \geq -3 \).
This means any value of \( x \) that is \( -3 \) or greater is acceptable in this expression, ensuring the value inside the square root is not negative. Inequalities are also about keeping the function valid and correctly defined.

Function Evaluation

Function evaluation refers to the process of finding the value of a function given a specific input. This is particularly useful when checking specific outputs of a function or when required to see the immediate result of substituting a specific \( x \) value.
In the context of the given exercise, the function \( f(x) = \sqrt{x+3} \) is evaluated by substituting \( x = 6 \).
Step-by-step:

• Substitute \( 6 \) into the function: \( \sqrt{6+3} \)
• Perform the arithmetic inside the square root: \( 6+3 = 9 \).
• Take the square root: \( \sqrt{9} = 3 \).

This process not only helps verify the function's value at that particular point but also serves to ensure the function's operation aligns with its defined rules and domains. It is a foundational concept in understanding how functions behave across different input values.