Problem 29 - Algebra 1

Question

Evaluate the function for the given value of \(x .\) Round your answer to the nearest tenth. $$y=\sqrt{36 x-2} ; \frac{1}{2}$$

Step-by-Step Solution

Answer

The value of \(y\) when \(x = \frac{1}{2}\) is 4.

Step 1: Plug in the Given Value of \(x\)

To begin, substitute \(x=\frac{1}{2}\) into the function, yielding \(y=\sqrt{36(\frac{1}{2})-2}\). This is the first step towards finding the value of \(y\).

Step 2: Simplify the Expression

Simplify inside the square root and the function becomes \(y=\sqrt{36*(\frac{1}{2})-2} = \sqrt{18-2} = \sqrt{16}\).

Step 3: Evaluate the Square Root

The number inside the square root sign, called the radicand, is a perfect square (16), which simplifies to 4. Thus, \(y = 4\).

Key Concepts

Square Root

The concept of the square root is essential in mathematics. It involves finding a number which, when multiplied by itself, gives the original number.
For instance, the square root of 16 is 4 because 4 times 4 equals 16.
In mathematical notation, the square root is represented by the radical symbol \( \sqrt{} \). When evaluating expressions with square roots, it’s important to recognize the radicand, which is the number under the square root sign.
Using our example, \( \sqrt{16} = 4 \). If the radicand is not a perfect square, you may need to approximate its square root, especially when rounding to a particular decimal place, such as the nearest tenth.

Simplifying Expressions

Simplifying expressions is a crucial step in solving many mathematical problems. It involves reducing an expression to its simplest form. This often means performing operations like addition, subtraction, multiplication, or division where possible.
An example is the expression inside the square root: \( 36 \times \frac{1}{2} - 2 \). This demonstrates multiple steps:

• First, you calculate \(36 \times \frac{1}{2}\), which simplifies to 18.
• Then, subtract 2 to get 16, which makes the expression under the square root symbol as simple as possible.

This simplification is vital before applying further operations, such as evaluating a square root.

Substitution Method

The substitution method is a technique used to solve equations by replacing one variable with a given value. This is a straightforward step often used to evaluate functions.
For the function \( y = \sqrt{36x - 2} \), and given \( x = \frac{1}{2} \), plug this value into the equation.
This gives you a new expression: \( y = \sqrt{36(\frac{1}{2}) - 2} \). With substitution, you essentially format the equation specifically for the substituted value, allowing you to calculate exact results.
Make sure the substitution is accurate to avoid errors, and simplify as much as possible following substitution to smoothly find your final answer.