Problem 28 - Algebra 1: Concepts and Skills

Question

Evaluate the function for the given value of x. $$y=\sqrt{21-2 x}:-2$$

Step-by-Step Solution

Answer

The value of y when x is -2 in the function \(y=\sqrt{21-2x}\) is 5, because \(\sqrt{25}\) equals 5.

Step 1: Identify and substitute the value of x

The given function is \(y=\sqrt{21-2x}\) and the given value of x is -2. This value of x needs to be substituted in the function.

Step 2: Apply the order of operations

Once the value of x is substituted into the function, apply the order of operations as follows. First, perform the operation in parenthesis, which is the multiplication of 2 and -2. After that, proceed to add or subtract (21 minus the result from this calculation), and then take the square root of that result.

Step 3: Simplify the expression

The expression now reads as \(y=\sqrt{21-(-4)}\). This then simplifies to \(y=\sqrt{21+4}=\sqrt{25}\).

Key Concepts

Square Root

The square root function is a mathematical operation that determines what number, when multiplied by itself, will yield the original number. Essentially, it asks, "What number squared equals this?"

• For example, the square root of 25 is 5, because 5 squared (5×5) is 25.
• The square root is represented by the symbol \( \sqrt{} \).
• It is vital to understand that there are usually two square roots for any positive number: a positive and a negative. However, when dealing with square roots in elementary calculus, the positive is often used by default. This positive square root is known as the principal square root.

When evaluating expressions with square roots, like \( y = \sqrt{21-2x} \), once you've simplified inside the square root, you can then find its square root. Here, after simplifying inside, the expression becomes \( \sqrt{25} \), which equals 5.

Order of Operations

The order of operations is a fundamental principle in mathematics that dictates the correct sequence to follow when evaluating expressions. This ensures consistent results across calculations. Remembering the order can be easily accomplished with the PEMDAS acronym:

• Parentheses
• Exponents (or square roots)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In the given problem, \( y = \sqrt{21-2x} \), once \( x = -2 \) is substituted, the subsequent steps adhere to PEMDAS:- Perform the multiplication of \( 2 \times -2 \).- Proceed with addition or subtraction inside the square root \( 21 - (-4) \).- Finally, take the square root of the remaining value, simplifying to \( \sqrt{25} \). Applying this order ensures that each part of the expression is evaluated accurately.

Substitution in Algebra

Substitution is a key technique in algebra used to simplify and solve expressions and equations. This process involves replacing variables with numeric values.In the exercise, we begin with the function \( y = \sqrt{21-2x} \). The task asks for the evaluation at a specific value, \( x = -2 \). By substituting \( -2 \) for \( x \), the function becomes a numerical expression: \( y = \sqrt{21-2(-2)} \).

• Substitution allows for simplification into a single numeric result.
• This step translates variables into tangible numbers, making complex expressions easier to handle.

Once the substitution is complete, further simplification and application of the order of operations can follow. This practice is essential for solving and understanding algebraic functions in greater depth.