Problem 50 - Algebra 2

Question

Find each value if \(f(x)=4 x+3\) and \(g(x)=5 x-7\). $$ f(-2) $$

Step-by-Step Solution

Answer

The value of \(f(-2)\) is -5.

Step 1: Understand the Problem

We need to evaluate the function \(f(x) = 4x + 3\) at \(x = -2\). This means that we should substitute \(-2\) for \(x\) in the expression \(4x + 3\).

Step 2: Substitute the Given Value

Replace \(x\) with \(-2\) in the function \(f(x) = 4x + 3\). This gives us: \(f(-2) = 4(-2) + 3\).

Step 3: Perform the Multiplication

Calculate \(4(-2)\). Since multiplying 4 by -2 equals -8, we have: \(f(-2) = -8 + 3\).

Step 4: Perform the Addition

Add \(-8\) and 3. \(-8 + 3 = -5\). So, \(f(-2) = -5\).

Step 5: Conclude the Solution

The evaluated result of the function \(f(x)\) at \(x = -2\) is \(-5\).

Key Concepts

Substitution

In mathematics, the concept of substitution is fundamental when working with functions. It involves replacing a variable with a specific value. The goal is to simplify the expression or to find the function's output at that point. For example, if you have a function \( f(x) = 4x + 3 \), to evaluate \( f(-2) \), you substitute \(-2\) for every \(x\) in the equation. This transforms the expression to \(4(-2) + 3\).
Substitution helps in logical problem-solving and is widely used in various aspects like solving equations, integrating functions, and evaluating expressions. By substituting values, you directly see how an input affects the function's output, which is crucial for understanding the behavior of functions.

Linear Functions

Linear functions are one of the simplest yet most important types of functions. A linear function can be represented in the form \( f(x) = mx + b \), where \(m\) is the slope and \(b\) is the y-intercept. These functions are graphed as straight lines on a coordinate plane, which is where they get their name, 'linear'.
The function \( f(x) = 4x + 3 \) is a perfect example of a linear function, where \( m = 4 \) determines the steepness of the line, and \( b = 3 \) indicates where the line crosses the y-axis. Evaluating linear functions for specific values like \(-2\) can help us understand the input-output relationship easily, as the operations involved (multiplication and addition) follow a straightforward pattern.

Arithmetic Operations

Arithmetic operations are the basic computations that form the foundation of most mathematical calculations, including those in functions. The basic operations include addition, subtraction, multiplication, and division.
In the example of evaluating the function \( f(x) = 4x + 3 \) at \(x = -2\), arithmetic operations come into play immediately after substitution. The first step involves multiplication: calculating \(4(-2)\), which results in \(-8\).
Subsequently, addition is performed to combine \(-8\) and 3, resulting in \(-5\). Each step in function evaluation involves one or more arithmetic operations, making an understanding of these simple but crucial operations vital for tackling more complex math problems. They help maintain the arithmetic sequence and logic, ensuring that each calculation leads correctly to the next.