Problem 26 - Algebra 1

Question

Evaluate the function for the given value of \(x .\) Round your answer to the nearest tenth. $$y=\sqrt{\frac{x}{2}-2} ; 22$$

Step-by-Step Solution

Answer

The output of the function when \(x\) is 22 is 3

Step 1: Substitute the given value into the function

First, replace every occurrence of \(x\) with 22 in the given function. This gives: \(y=\sqrt{\frac{22}{2}-2}\)

Step 2: Simplify the expression

After the substitution, the function becomes \(y=\sqrt{\frac{22}{2}-2} = \sqrt{11-2} = \sqrt{9}\)

Step 3: Find the square root

The square root of 9 is 3, so we have \(y = 3\)

Key Concepts

Square Root

The concept of a square root is fundamental in mathematics and involves finding a number that, when multiplied by itself, equals the original number. For example, the square root of 9 gives 3, because 3 multiplied by 3 equals 9.

When dealing with functions involving square roots, it's important to understand how they affect the meaning of an expression. In this context, the square root function provides a non-negative output, also known as the principal square root.

• If an expression under a square root becomes negative, it usually precludes real solutions, since square roots of negative numbers fall into the realm of complex numbers.
• It's essential to simplify under the square root before calculating it to avoid unnecessary complexity.

Being comfortable with square roots can help unravel many mathematical problems quickly and efficiently.

Simplification

Simplification in mathematics is all about reducing expressions to their simplest form. This often involves combining like terms, reducing fractions, and performing arithmetic operations.

• The end goal is to make an expression as easy to work with as possible, eliminating any unnecessary complexity.
• For instance, transforming the fraction \( \frac{22}{2} \) directly simplifies to 11, making it much easier to proceed with further calculations.

In the provided function, simplification was key in transforming the expression inside the square root from \( \frac{22}{2} - 2 \) to \( 11 - 2 \), resulting in \( \sqrt{9} \). Simplification helps in evaluating expressions swiftly and accurately by making numbers easier to manage.

Expression Substitution

Expression substitution is a method used in mathematics to replace variables in functions with specific values. This approach allows us to evaluate or transform functions based on given conditions or inputs.

• This substitution process forms the backbone of solving many algebraic expressions and equations.
• For example, substituting 22 for \( x \) in the function \( y = \sqrt{\frac{x}{2}-2} \) changes the function to \( y = \sqrt{\frac{22}{2}-2} \).

Proper substitution is crucial to ensure that calculations are based on the correct inputs, allowing for accurate outcomes. By substituting correctly, the function in the provided example could be simplified and evaluated, providing a precise solution.