Problem 74 - Algebra 2

Question

Find each function value. \(f(x)=3 x-1, f\left(\frac{1}{2}\right)\)

Step-by-Step Solution

Answer

The function value is \( \frac{1}{2} \).

Step 1: Understand the Problem

We are given the function \( f(x) = 3x - 1 \) and asked to find the value of \( f \left( \frac{1}{2} \right) \). This means we need to substitute \( \frac{1}{2} \) for \( x \) in the function.

Step 2: Substitute the Value of x

Replace \( x \) in the function with \( \frac{1}{2} \). The expression becomes:\[ f \left( \frac{1}{2} \right) = 3 \times \frac{1}{2} - 1 \]

Step 3: Perform the Multiplication

Calculate the product \( 3 \times \frac{1}{2} \):\[ 3 \times \frac{1}{2} = \frac{3}{2} \]

Step 4: Subtraction

Subtract 1 from \( \frac{3}{2} \). We need to express 1 as a fraction with the same denominator:\[ 1 = \frac{2}{2} \]Now subtract:\[ \frac{3}{2} - \frac{2}{2} = \frac{1}{2} \]

Step 5: Conclusion

The value of the function \( f(x) = 3x - 1 \) when \( x = \frac{1}{2} \) is \( \frac{1}{2} \).

Key Concepts

Algebraic Functions

Algebraic functions are expressions constructed using algebraic operations such as addition, subtraction, multiplication, division, and raising to a power. They involve variables and constants. In the given exercise, the function is defined as \( f(x) = 3x - 1 \), where \( x \) is the variable. Algebraic functions are essential as they help us describe various mathematical relationships.

• Understanding Variables: The variable \( x \) is a placeholder that can take different values. In the function \( f(x) = 3x - 1 \), the function value changes according to the value of \( x \).
• Constructing Functions: Algebraic functions include operations like multiplying \( 3 \) by \( x \) and then subtracting \( 1 \), as seen here. The formula gives a straightforward way to apply these operations to any value of \( x \).

When you work with algebraic functions, you learn to manipulate these expressions to find specific values.

Substitution Method

The substitution method is a technique used in mathematics to simplify expressions or equations, particularly for evaluating functions. Here, replacing \( x \) with a specific value helps find that particular function value. For the function \( f(x) = 3x - 1 \), the task involves calculating \( f\left(\frac{1}{2}\right)\).

• Understanding Substitution: Start by replacing the variable \( x \) with the given value, \( \frac{1}{2} \). This changes the original expression into one you can evaluate.
• Applying to the Example: Substituting \( \frac{1}{2} \) into the function gives \( 3 \times \frac{1}{2} - 1 \).

After substitution, subsequent simplifications are needed to complete the evaluation.

Fraction Operations

Fraction operations involve arithmetic manipulation of fractions, such as addition, subtraction, multiplication, and division. These operations frequently appear in function evaluations, as with the task of finding \( f\left(\frac{1}{2}\right) \).

• Performing Multiplication: For the expression \( 3 \times \frac{1}{2} \), multiply the whole number by the fraction: \( 3 \times \frac{1}{2} = \frac{3}{2} \).
• Executing Subtraction: To subtract \( 1 \) from \( \frac{3}{2} \), convert \( 1 \) into a fraction: \( 1 = \frac{2}{2} \), then perform \( \frac{3}{2} - \frac{2}{2} = \frac{1}{2} \).

Mastering fraction operations is key in evaluations, as they often transform difficult expressions into understandable results.