Problem 22 - Algebra 1

Question

Evaluate the function for the given value of \(x .\) Round your answer to the nearest tenth. $$y=\sqrt{x-7} ; 15$$

Step-by-Step Solution

Answer

The function \(y = \sqrt{x-7}\) evaluated for \(x = 15\), rounded to the nearest tenth, equals 2.8

Step 1: Substitute the given value of x

The given value of \(x\) is 15. Substitute this value into the function. Thus, the modified function becomes \(y = \sqrt{15-7}\)

Step 2: Simplify the expression inside the square root

Inside the square root is the expression 15 - 7 which simplifies to 8. Thus, the function can be rewritten as \(y = \sqrt{8}\).

Step 3: Calculate the square root

The value of \(\sqrt{8}\) can be calculated using a calculator. The resulting value is approximately 2.828427125.

Step 4: Round the answer to the nearest tenth

The result from the previous step is rounded to the nearest tenth as requested. The decimal 2.828427125 rounded to the nearest tenth is 2.8.

Key Concepts

Understanding Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because

• 3 × 3 = 9

Square roots are often expressed using the radical symbol “\( \sqrt{} \)”. For example, \( \sqrt{8} \) is asking "what number times itself equals 8?".

Calculating square roots by hand can be tricky, so it's often done with a calculator, especially for numbers like 8, which don’t have an exact whole number square root. In the original exercise, we find that \( \sqrt{8} \) is approximately 2.828427125.

Rounding Numbers for Simplicity

Rounding numbers is a way to simplify them. This makes them easier to work with or understand. In mathematics, rounding involves adjusting a number to its nearest whole number, tenth, hundredth, etc.

To round to the nearest tenth, we look at the number in the hundredths place:

• If it is 5 or more, round up.
• If it is less than 5, round down.

For example, rounding 2.828427125 to the nearest tenth, focus on the hundredths digit (2 in this case). Since it's less than 5, we round down to 2.8.

Substitution in Function Evaluation

Substitution in mathematics means replacing a variable with a given value. It's a key step in evaluating functions. If you have a function like \( y = \sqrt{x - 7} \), you substitute any given \(x\) value into the function.

For example:

• Suppose \( x = 15 \), substitute it into the equation: \( y = \sqrt{15 - 7} \).
• This simplifies to \( y = \sqrt{8} \).

Substitution helps you find specific answers and evaluate how changes in \(x\) affect the function.