Problem 22 - Algebra 1: Concepts and Skills

Question

Evaluate the function for the given value of x. $$y=\sqrt{32 x} ; 2$$

Step-by-Step Solution

Answer

The value of the function \(y = \sqrt{32x}\) at the point \(x = 2\) is 8.

Step 1: Substitute x into the function

To find the value of the function \(y\) at the point \(x = 2\), substitute \(2\) into the equation wherever \(x\) appears. This gives us: \(y = \sqrt{32*2}\) = \( \sqrt{64} \).

Step 2: Simplify the Square Root

The next step is to simplify the square root. The square root of 64 is 8. So, \(y = 8\).

Key Concepts

Square Roots

Square roots are fundamental in mathematics and are often encountered when dealing with quadratic equations and functions. A square root asks the question: what number, when multiplied by itself, results in the given number? For example, the square root of 64, denoted as \( \sqrt{64} \), is 8 because \( 8 \times 8 = 64 \).

When you encounter a square root in an expression, the goal is usually to simplify it by finding the positive number that, when squared, returns to the original value under the square root symbol. These roots are useful in various calculations, especially when dealing with geometrical problems or physics equations.

• Remember: the square root of zero is always zero.
• The square root is only defined for non-negative numbers in the realm of real numbers.

Working with square roots helps in finding precision in areas like statistics, probability, and many sciences.

Function Substitution

Function substitution is a basic operation in algebra where you replace a variable with a given number. This is essential for finding specific values of functions at certain inputs. For instance, given \( y = \sqrt{32x} \) and \( x = 2 \), you substitute 2 for every instance of \( x \).

This substitution turns \( y = \sqrt{32x} \) into \( y = \sqrt{32 \times 2} \) which simplifies further. Function substitution is crucial because it allows us to evaluate or solve functions for specific inputs, thus providing clear and concrete results.

• Always ensure the value you're substituting fits the domain of the function.
• Double-check your substitution to make sure all instances of the variable have been replaced.

By practicing substitution, one gains better intuition for handling equations and understanding functional relationships.

Simplifying Expressions

Simplifying expressions is a central skill in algebra. It involves reducing expressions into their most basic form while keeping their value intact. This skill is critical in solving equations efficiently and accurately.

In our example, after substituting \( x = 2 \) into \( y = \sqrt{32x} \), you are left with \( y = \sqrt{64} \). Simplifying that involves recognizing that 64 is a perfect square, so \( \sqrt{64} = 8 \).

• Look for common factors or patterns in numbers to simplify quicker.
• Always aim to express answers in the simplest form possible for clarity.

Simplifying helps in revealing the true nature of mathematical expressions and makes them more manageable for further operations.