Problem 32 - Algebra and Trigonometry

Question

Use the function to evaluate the indicated expressions and simplify. $$ f(x)=3 x-1 ; \quad f(2 x), 2 f(x) $$

Step-by-Step Solution

Answer

\( f(2x) = 6x - 1 \), and \( 2f(x) = 6x - 2 \).

Step 1: Evaluate f(2x)

Given the function \( f(x) = 3x - 1 \), we need to find \( f(2x) \). Substitute \( 2x \) in place of \( x \) in the function. Thus, \( f(2x) = 3(2x) - 1 = 6x - 1 \).

Step 2: Evaluate 2f(x)

Evaluate \( 2f(x) \) using the given function \( f(x) = 3x - 1 \). This means finding \( 2 \times (3x - 1) \), which simplifies to: \( 2f(x) = 2(3x - 1) = 6x - 2 \).

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental concept in mathematics that combines numbers, variables, and operations like addition or multiplication to represent a value.For instance, in the function \( f(x) = 3x - 1 \), "3x - 1" is an algebraic expression.

• **Variables**: Letters like \( x \) represent numbers that can change or vary.
• **Coefficients**: Numbers like 3, which multiply the variables.
• **Constants**: Numbers without variables, like -1, which are fixed.

Algebraic expressions are versatile and can model real-world situations from physics problems to economics.Understanding them is crucial as they form the backbone of solving equations and functions.When you analyze an expression, pay close attention to each component as this will guide you in manipulation and evaluation tasks.

Substitution in Functions

Substitution is a powerful tool in algebra, especially when dealing with functions.Essentially, substitution involves replacing a variable in a function with another value or expression.In the exercise, we see substitution in action when finding \( f(2x) \) from the function \( f(x) = 3x - 1 \).You replace \( x \) with \( 2x \) in the expression. The new expression now becomes \( f(2x) = 3(2x) - 1 \), which simplifies to \( 6x - 1 \).

• **Step-by-step substitution**: Start by writing the expression as it is, but replace every occurrence of the variable with the substitution value.
• **Keep track of operations**: Ensure you apply each operation (like multiplication) accurately when substituting.

Substitution allows you to explore how changes in variables affect the overall function, providing a deeper insight into its behavior and resulting values.

Simplifying Algebraic Expressions

Simplifying an algebraic expression involves combining like terms and making the expression as concise as possible.In our example, simplifying is demonstrated when evaluating \( 2f(x) \).From \( 2f(x) = 2(3x - 1) \), this breaks down to \( 2 \times 3x \) and \( 2 \times -1 \), leading to \( 6x - 2 \).

• **Combining like terms**: Terms that have the same variable raised to the same power can be combined. For example, \( 3x + 4x \) becomes \( 7x \).
• **Distributive Property**: This is key in simplifying expressions with parentheses, like \( 2(3x - 1) \), where the 2 multiplies each term inside.

The goal of simplification is to make calculations easier and the expression clearer. This process is foundational for solving equations as it standardizes expressions, making comparisons and further manipulations more straightforward.