Problem 63 - Algebra 2

Question

Find each value if \(f(x)=3 x-4\). $$ f(a) $$

Step-by-Step Solution

Answer

\( f(a) = 3a - 4 \)

Step 1: Understand the Function Notation

The function given is \( f(x) = 3x - 4 \). This means for whatever value is substituted for \( x \) in \( f \), we compute it using the expression \( 3x - 4 \). Here, we need to find \( f(a) \), so we'll substitute \( a \) into the function.

Step 2: Substitute the Value in the Expression

Replace \( x \) with \( a \) in the function \( f(x) = 3x - 4 \) to find \( f(a) \). This gives the expression \( f(a) = 3a - 4 \).

Step 3: Simplify the Expression

The expression \( 3a - 4 \) is already in its simplest form because it combines all the operations possible with the given variable \( a \). Thus, \( f(a) = 3a - 4 \) is the simplified result and the value of the function at \( a \).

Key Concepts

Function Evaluation

Function evaluation is the process of finding the output of a function given a specific input. When you have a function, like \( f(x) = 3x - 4 \), it's essentially a rule that tells you how to transform an input \( x \) into an output. Evaluating a function involves substituting a number or expression for the variable in the function's formula.

The process is quite straightforward:

• Identify the input value you need to substitute. In this case, we were asked to find \( f(a) \).
• Replace the variable in the function's formula with the chosen input value. Therefore, \( f(a) = 3a - 4 \).
• Compute the result using the arithmetic operations defined in the function. This will provide the output for the given input.

Understanding function evaluation is crucial as it lays the foundation for solving more complex problems in algebra and calculus.

Algebraic Expression

An algebraic expression is a mathematical phrase that includes numbers, variables, and operations like addition, subtraction, multiplication, and division. These expressions represent quantities and are used to express a mathematical relationship.

In our example, the function expression \( 3x - 4 \) is algebraic. It comprises:

• The coefficient 3, which multiplies the variable \( x \). It denotes that \( x \) is scaled by 3.
• The variable \( x \), which is a placeholder that can represent various values.
• The constant -4, which is subtracted from the product of 3 and \( x \).

This simple expression follows the rules of algebra, and understanding it helps make sense of more complex mathematical problems by breaking them down into manageable parts.

Variable Substitution

Variable substitution involves replacing a variable in an expression or formula with a specific value or another expression. This concept is vital for evaluating functions, simplifying expressions, and solving equations.

In function notation, substituting the value involves:

• Identifying the variable to substitute. In our example, it's \( x \).
• Replacing this variable with the desired value, such as \( a \) in \( f(a) = 3a - 4 \).
• Ensuring that the substitution is done consistently throughout the expression.

Through substitution, you can transform a general expression into a specific result, making calculations possible. Mastering this process is vital for tackling a wide variety of mathematical tasks.