Problem 21 - Algebra 1

Question

Evaluate the function for the given value of \(x .\) Round your answer to the nearest tenth. $$y=\frac{1}{2} \sqrt{x}-1 ; 16$$

Step-by-Step Solution

Answer

The solution for \(y\) when \(x = 16\) in the given function is 1.

Step 1: Substitute x into the function

Replace \(x\) with 16 in the function \(y = \frac{1}{2} \sqrt{x} - 1\). So, we have \(y = \frac{1}{2} \sqrt{16} - 1\)

Step 2: Simplify Root

Calculate the square root of 16 which equals 4. Substituting this back in gives \(y = \frac{1}{2} \times 4 - 1\)

Step 3: Perform Multiplication

Perform the multiplication operation first (according to Order of Operations), so we have \(y = 2 - 1\)

Step 4: Perform Subtraction

Perform the subtraction operation to obtain \(y = 1\)

Step 5: Round to the Nearest 10th

In this case, rounding to the nearest tenth does not change our answer, so \(y = 1\).

Key Concepts

Square Root

The square root is a fundamental concept in mathematics, representing a number that, when multiplied by itself, gives the original number. In the exercise, the square root operation appears as \( \sqrt{x} \), focusing on finding the square root of 16. The square root of 16 is 4, because 4 times 4 equals 16.

• Square root is denoted by the symbol \( \sqrt{} \).
• It is the opposite operation of squaring a number.
• For numbers like 16, which are perfect squares, finding the square root is straightforward.

Remember, calculating the square root is crucial when simplifying expressions like \( \frac{1}{2} \sqrt{x} - 1 \), as seen in our example. Precise operations ensure accuracy in reaching the final solution when working with functions.

Order of Operations

Order of operations is a set of rules that determines the sequence in which mathematical operations are performed. It ensures the calculations are done consistently and correctly. In our problem, using the order of operations correctly is key:

• Perform any calculations inside parentheses or brackets first.
• Next, calculate exponents and roots, like square roots.
• Then, carry out multiplication and division, moving from left to right.
• Finally, perform addition and subtraction from left to right.

In the example \( y = \frac{1}{2} \sqrt{16} - 1 \), first, we find the square root of 16. Next, we multiply by \( \frac{1}{2} \), followed by subtraction. This process guarantees the expression is simplified correctly to obtain the accurate result.

Substitution

Substitution in math involves replacing variables with numbers or other expressions. In our function \( y = \frac{1}{2} \sqrt{x} - 1 \), substituting \( x \) with 16 facilitates the evaluation of the function.

• Identify the variable to substitute.
• Replace the variable with the given value.
• Simplify the expression to find the result.

Here, substituting allows finding \( y \) by clearing \( x \) and focusing on computations with concrete numbers. Substitution is essential in evaluating functions, making them versatile and practical for various inputs. It simplifies the process of moving from general formulas to specific solutions.