Problem 63 - Elementary and Intermediate Algebra

Question

Find each function value, if possible. Do not use a calculator. $$f(x)=\sqrt{3 x+1}$$ a. \(f(8)\) b. \(f(-2)\)

Step-by-Step Solution

Answer

\(f(8) = 5\), but \(f(-2)\) is not a real number.

Step 1: Substitute the Value

To find \(f(8)\), substitute 8 into the function: \(f(8) = \sqrt{3 \cdot 8 + 1}\).

Step 2: Calculate Inside the Square Root

Calculate the expression inside the square root: \(3 \cdot 8 + 1 = 24 + 1 = 25\).

Step 3: Find the Function Value

Take the square root of 25 to find \(f(8)\): \(\sqrt{25} = 5\). Therefore, \(f(8) = 5\).

Step 4: Substitute Another Value

To find \(f(-2)\), substitute -2 into the function: \(f(-2) = \sqrt{3 \cdot (-2) + 1}\).

Step 5: Calculate Inside the Square Root Again

Calculate the expression inside the square root for \(f(-2)\): \(3 \cdot (-2) + 1 = -6 + 1 = -5\).

Step 6: Check for Real Number Result

Because the expression inside the square root is negative (-5), the function \(f(-2)\) does not have a real number value, since square roots of negative numbers are not defined within the real numbers.

Key Concepts

Square Root Function

Square root functions are an important part of mathematics, allowing us to understand how numbers relate to their respective square roots. A square root function is typically written as \( f(x) = \sqrt{something} \). This function only yields real number results when the value under the square root (the radicand) is non-negative. This is because within the set of real numbers, a square root of a negative number is not defined.

• For example, \( f(x) = \sqrt{3x + 1} \) means we look at values of \( x \) that make the expression inside the square root non-negative.
• When you substitute a number into the function, say, \( f(8) \), you replace \( x \) with 8, evaluate the expression, and then determine the square root of the resulting number.
• Thus, \( f(8) = \sqrt{3 \cdot 8 + 1} = \sqrt{25} = 5 \).
• If you substitute a number that results in a negative radicand, like \( f(-2) \), the function does not yield a real number value.

Understanding square root functions is crucial for solving various algebraic problems and understanding further mathematical concepts.

Real Numbers

Real numbers include all the numbers on the number line and are divided into rational and irrational numbers. Rational numbers are numbers that can be expressed as fractions, such as 1/2, while irrational numbers cannot be expressed as exact fractions, such as \( \pi \) and \( \sqrt{2} \).

• Every number you can think of in everyday context, like whole numbers, fractions, and decimals, are all real numbers.
• They include both positive and negative numbers as well as zero.
• Real numbers are used in square root functions to determine if the radicand results in a real number when square rooted.

When evaluating a function like \( f(x) = \sqrt{3x + 1} \), finding \( f(x) \) is only possible when the expression inside the square root is non-negative, ensuring the output is a real number.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. Understanding algebraic expressions is critical when dealing with functions like the square root function.

• The expression \( 3x + 1 \) in the square root function \( f(x) = \sqrt{3x + 1} \) is an example of an algebraic expression.
• In working with algebraic expressions, substitution is a common technique where you replace a variable with a specific number to evaluate the expression.
• Following arithmetic operations is essential, ensuring you handle multiplication, addition, and any exponents in the correct order.

When substituting values into an algebraic expression, like in \( f(8) = \sqrt{3 \cdot 8 + 1} \), ensure you calculate each step carefully. This ensures a correct final result of the function.