Problem 7 - Algebra 1: Concepts and Skills

Question

Evaluate the function for x 0, 1, 2, 3, and 4. Round your answers to the nearest tenth. $$y=\sqrt{x+2}$$

Step-by-Step Solution

Answer

The function values for \(x=0, 1, 2, 3, 4\) are approximately \(1.4, 1.7, 2.0, 2.2, 2.4\) respectively when rounded to the nearest tenth.

Step 1: Substituting \(x = 0\)

Replace \(x\) in the function with 0, \(y = \sqrt{0 + 2}\)

Step 2: Substituting \(x = 1\)

Replace \(x\) in the function with 1, \(y = \sqrt{1 + 2}\)

Step 3: Substituting \(x = 2\)

Replace \(x\) in the function with 2, \(y = \sqrt{2 + 2}\)

Step 4: Substituting \(x = 3\)

Replace \(x\) in the function with 3, \(y = \sqrt{3 + 2}\)

Step 5: Substituting \(x = 4\)

Replace \(x\) in the function with 4, \(y = \sqrt{4 + 2}\)

Step 6: Calculating \(y\) values

Using the arithmetic operations of addition and square root, calculate the \(y\) value for each substitution.

Step 7: Rounding to the nearest tenth

Once the \(y\) value for each substitution is calculated, don't forget to round the results to the nearest tenth.

Key Concepts

Substitution Method

The substitution method is a simple and effective way to evaluate functions. It involves taking a specific value and replacing the variable in the function with that value. This allows you to find the output, or the result, of the function for those specific inputs. In our context, we substitute different values of \(x\) into the given function \(y = \sqrt{x+2}\) one at a time.

• First, you start with \(x = 0\), then substitute it into the function: \(y = \sqrt{0 + 2} = \sqrt{2}\).
• Next, try \(x = 1\) and follow the same process: \(y = \sqrt{1 + 2} = \sqrt{3}\).

This process is continued for each subsequent value of \(x\) (e.g., 2, 3, and 4). By substituting the variable, you are simplifying the problem and making it manageable to solve for each particular case.

Square Root Function

A square root function is a type of function denoted by the mathematical symbol \( \sqrt{} \). It evaluates to the number that, when multiplied by itself, returns the input number. In the equation \(y = \sqrt{x+2}\), you solve for \(y\) by finding the square root of the expression inside the square root symbol.

• The function inside the square root first adds 2 to your input value \(x\), which guides the overall shape and position of the curve.
• Once you perform this addition, you compute the square root of the result, giving you the corresponding \(y\) value.

Understanding square root functions allows you to determine the potential outputs for a variety of inputs as they will result in positive numbers since the square root of a negative is not defined in the real number system.

Rounding Numbers

Rounding numbers involves adjusting them to a specific degree of precision, which makes them easier to work with while maintaining accuracy. In this exercise, your task is to round each \(y\) value to the nearest tenth after calculating the square root.

• If the digit in the hundredths place (the second digit after the decimal) is 5 or greater, round up the tenths place by one.
• If it's less than 5, keep the tenths digit the same.

For instance, after calculating \(y = \sqrt{2} \approx 1.414\), you would round to 1.4. Rounding is an important skill, helping simplify results without significantly affecting the precision required for practical decision-making.