Problem 23 - Algebra 1: Concepts and Skills

Question

Evaluate the function for the given value of x. $$y=\sqrt{3 x} ; 12$$

Step-by-Step Solution

Answer

The value of the function \(y=\sqrt{3x}\) for \(x=12\) is 6.

Step 1: Understanding the Function

The given function is \(y=\sqrt{3x}\). This is a square root function, where the variable \(x\) is multiplied by 3 and then the square root is taken.

Step 2: Substitution

In the function \(y=\sqrt{3x}\), replace \(x\) with 12, the value given in the problem. The function then becomes \(y=\sqrt{3*12}\).

Step 3: Simplification

Simplify the expression under the root. Multiply 3 and 12 to get 36. So, the function becomes \(y=\sqrt{36}\). Calculate the square root of 36 to yield 6.

Key Concepts

Square Root Function

Let's dive into what a square root function is before evaluating our specific example. In mathematics, a square root function involves taking the square root of a variable or expression. The general form of a square root function is \(y = \sqrt{x}\). Square roots answer the question: "What number, when multiplied by itself, will give me this specific value?"

It's important to note that:

• The result of a square root is always a non-negative number, known as the principal square root.
• Square root functions have a domain where the value inside the square root is non-negative, which ensures the function is defined for real numbers.

In our problem, the function given is \(y=\sqrt{3x}\). This is a square root function of the form \(y = \sqrt{ax}\), where \(a = 3\). Understanding this helps us to proceed correctly in evaluating the function.

Function Evaluation

Evaluating a function means determining the output, or result, of the function for a given input. In our case, we are tasked with evaluating the function \(y = \sqrt{3x}\) when \(x = 12\).

To evaluate a function:

• Substitute the given value into the function in place of the variable. Here, replace \(x\) with 12.
• The function becomes \(y = \sqrt{3 \times 12}\).

This straightforward process allows us to determine the value of the function for any specified input.

Algebraic Simplification

Now that we have substituted \(x = 12\) into the function \(y = \sqrt{3x}\), the expression transforms to \(y = \sqrt{3 \times 12}\).

Simplification is a step where we make the expression easier to work with, by performing arithmetic operations. Here's how you can simplify this particular function:

• First, calculate the product inside the square root: \(3 \times 12 = 36\).
• Next, take the square root of 36. Since 36 is a perfect square, its square root is positive 6. Thus, \(y = \sqrt{36} = 6\).

Simplifying expressions clarifies the function's output, ultimately revealing our final answer of 6 when \(x = 12\).