Problem 5 - Algebra 1: Concepts and Skills

Question

Evaluate the function \(f(x)=-5 x-2\) for the given value of \(x\) $$ x=-2 $$

Step-by-Step Solution

Answer

The evaluated value of the function \(f(x) = -5x - 2\) for \(x = -2\) is 8.

Step 1: Identify the Function and the Value for Evaluation

The function to evaluate is \(f(x) = -5x - 2\) and the value for \(x\) is -2.

Step 2: Substitute x Value in the Function

Replace \(x\) in the function \(f(x) = -5x - 2\) with the given value of -2, leading to \(f(-2) = -5(-2) - 2\).

Step 3: Perform the Operation

Perform the multiplication and addition operations, resulting in \(f(-2) = 10 - 2\).

Step 4: Find Final Value

After performing the operation, the final value of the function becomes \(f(-2) = 8\).

Key Concepts

Understanding Linear Functions

Linear functions are an essential concept in algebra and are often represented in the equation form: \( f(x) = mx + b \), where \( m \) and \( b \) are constants. Here, \( m \) is the slope of the line, which shows how steep the line is in a graph, and \( b \) is the y-intercept, which is the point where the line crosses the y-axis.
Linear functions are called 'linear' because their graphs depict straight lines. These functions show a constant rate of change, meaning the function increases or decreases by the same amount for each unit change in the value of \( x \).

• Slope: Determines the direction and steepness of the line.
• Y-intercept: The starting value or point if no changes in \( x \) occur.
• Graph: Plots a straight line in coordinate planes.

Understanding this concept is fundamental because it is a building block for more advanced algebra and calculus topics.

Breaking Down Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (like \( x \) and \( y \)), and operations (such as addition, subtraction, multiplication, and division). They do not have an equality sign, which differentiates them from equations. Algebraic expressions allow us to generalize mathematical concepts and operations.
In the function \( f(x) = -5x - 2 \), the expression consists of

• Coefficient: \(-5\) is the coefficient of \( x \), showing how much \( x \) is multiplied by.
• Constant Term: \(-2\), which is added to or subtracted from the product of \( x \) and its coefficient.

Algebraic expressions can be simplified or evaluated for specific values, making them versatile tools in solving mathematical problems and real-world scenarios.

Utilizing the Substitution Method

The substitution method is a straightforward process used to evaluate expressions or solve equations by replacing variables with given numerical values. This technique is extremely useful when you want to determine the outcome of a function or verify solutions.
In our example, the function \( f(x) = -5x - 2 \) needs to be evaluated at \( x = -2 \). Using the substitution method:

• Identify: The variable(s) to substitute, here \( x = -2 \).
• Substitute: Replace \( x \) with the value \(-2\) in the function: \(-5(-2) - 2\).
• Calculate: Solve the expression step-by-step to find the result \( f(-2) = 10 - 2 = 8 \).

This method simplifies the evaluation process and helps verify that calculations are correct, whether confirming solutions or tackling complex algebraic problems.