Problem 8 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The function \(y=\sqrt{4x-1}\) cannot be evaluated for x = 0 in real numbers. For the other given values, after rounding to the nearest tenth, the function yields: For x=1, y=1.7; for x=2, y=2.6; for x=3, y=3.3; and for x=4, y=3.9.

Step 1: Evaluate the Function for x = 0

Substitute x = 0 into the function \(y=\sqrt{4x-1}\), so it becomes \(y=\sqrt{4(0) - 1}\), which simplifies to \(y = \sqrt{-1}\). However, the square root of a negative number is not possible in real numbers.

Step 2: Evaluate the Function for x = 1

Substitute x = 1 into the function \(y=\sqrt{4x-1}\), so it becomes \(y=\sqrt{4(1) - 1}\), which simplifies to \(y = \sqrt{3}\). After rounding to the nearest tenth, the value of y is approximately 1.7.

Step 3: Evaluate the Function for x = 2

Substitute x = 2 into the function \(y=\sqrt{4x-1}\), so it becomes \(y=\sqrt{4(2) - 1}\), which simplifies to \(y = \sqrt{7}\). After rounding to the nearest tenth, the value of y is approximately 2.6.

Step 4: Evaluate the Function for x = 3

Substitute x = 3 into the function \(y=\sqrt{4x-1}\), so it becomes \(y=\sqrt{4(3) - 1}\), which simplifies to \(y = \sqrt{11}\). After rounding to the nearest tenth, the value of y is approximately 3.3.

Step 5: Evaluate the Function for x = 4

Substitute x = 4 into the function \(y=\sqrt{4x-1}\), so it becomes \(y=\sqrt{4(4) - 1}\), which simplifies to \(y = \sqrt{15}\). After rounding to the nearest tenth, the value of y is approximately 3.9.

Key Concepts

Square Roots

A square root is a value that, when multiplied by itself, gives the original number. For example, the square root of 4 is 2, because 2 multiplied by 2 equals 4. The symbol for square root is \(\sqrt{}\). When you see \(\sqrt{n}\), you are looking for the number that, squared (or raised to the power of 2), equals \(n\). In the equation \(y=\sqrt{4x-1}\), you calculate a square root by substituting various values for \(x\) and simplifying the expression underneath the square root symbol.

• If the result inside the square root is negative, as in the case of \(x=0\) where \(\sqrt{-1}\) appears, the square root is not defined in the set of real numbers.
• Square roots can result in irrational numbers, meaning they cannot be expressed accurately as simple fractions (e.g., \(\sqrt{3}\)). These are often rounded or approximated for practical use.

Domain of a Function

The domain of a function is the set of all possible input values \(x\) that the function can accept. For the function \(y=\sqrt{4x-1}\), the expression under the square root \((4x-1)\) must be greater than or equal to zero, because real square roots of negative numbers do not exist. To find the domain:

• Set the expression inside the square root \(4x-1 \geq 0\).
• Solve the inequality: \(4x \geq 1\).
• Divide both sides by 4 to isolate \(x\): \(x \geq \frac{1}{4}\).

Thus, the domain of the function is all real numbers \(x\) such that \(x \geq \frac{1}{4}\). This means that the smallest value \(x\) can take is \(\frac{1}{4}\), and it can take any number larger than this.

Rounding Numbers

Rounding numbers is a way to simplify them, making them easier to work with without losing significant accuracy for practical use. When evaluating the function \(y=\sqrt{4x-1}\), the results are often rounded to a specific decimal place. Here, the directive was to round to the nearest tenth.

• The nearest tenth requires looking at the first digit after the decimal point. Determine if this digit will stay the same or if it will round up. To do this, check the digit in the hundredths place (second digit after the decimal).
• If this digit is 5 or greater, increase the tenths digit by 1. If it's less than 5, the tenths digit stays the same.

The practical examples in this exercise demonstrate that \(\sqrt{3}\) is rounded to 1.7, \(\sqrt{7}\) to 2.6, \(\sqrt{11}\) to 3.3, and \(\sqrt{15}\) to 3.9. This kind of rounding brings consistency and helps in approximating values when exact numbers are not necessary.