Problem 85 - Algebra 2

Question

Let \(f(x)=-4 x+1\) and \(g(x)=2 x-6 .\) Find \((g-f)(x)\) $$\begin{array}{llll}{\text { A. } 6 x-5} & {\text { B. } 6 x-7} & {\text { C. }-6 x+5} & {\text { D. }-6 x+7}\end{array}$$

Step-by-Step Solution

Answer

The answer is B: \(6x - 7\).

Step 1: Identify the functions

Function \(f(x)\) is given as \(-4x + 1\) and function \(g(x)\) as \(2x - 6\).

Step 2: Calculate \((g-f)(x)\)

The calculation \((g-f)(x)\) means \(g(x) - f(x)\), which results in: \( (2x - 6) - (-4x + 1)\).

Step 3: Simplify the expression

By simplifying the expression, it becomes: \(2x - 6 + 4x - 1\). When we combine like terms, this results in: \(6x - 7\)

Key Concepts

Linear Functions

Linear functions are equations that make a straight line when plotted on a graph. This type of function is one of the simplest forms of algebraic equations. The standard form of a linear function is expressed as \(f(x) = mx + b\), where:

• \(m\) is the slope of the line.
• \(b\) is the y-intercept, where the line crosses the y-axis.

Using our example, the linear function \(f(x) = -4x + 1\) has a slope of \(-4\) and a y-intercept of \(1\). This tells us the line decreases as \(x\) increases. Similarly, \(g(x) = 2x - 6\) has a slope of \(2\) with a y-intercept of \(-6\), indicating the line increases as \(x\) increases. Understanding the slope and y-intercept helps you picture how the line behaves, making it easier to perform operations like subtraction or addition of functions.

Function Subtraction

Function subtraction involves finding the difference between two functions. Given functions \(f(x)\) and \(g(x)\), the subtraction \((g-f)(x)\) is calculated as \(g(x) - f(x)\). This operation is straightforward yet essential for combining functions by subtracting their outputs.Let’s walk through our practice example. We were given \(g(x) = 2x - 6\) and \(f(x) = -4x + 1\) and asked to find \((g-f)(x)\). To compute it, take each corresponding part from both functions:

• Subtract the constant and variable parts of \(f(x)\) from \(g(x)\).
• The resulting expression becomes: \(2x - 6 - (-4x + 1)\).

In this way, subtracting functions is akin to subtracting regular numbers but you keep track of the coefficients of \(x\) and constant terms independently.

Expression Simplification

Expression simplification is a process of reducing an algebraic expression to its simplest form. After subtracting two functions, you often need to simplify the resulting expression to understand its behavior better.In our example, the subtraction \((g-f)(x)\) resulted in \(2x - 6 - (-4x + 1)\). Simplifying is achieved by removing parentheses and combining like terms:

• First, distribute any negative signs so \(-(-4x + 1)\) becomes \(+4x - 1\).
• The expression becomes \(2x - 6 + 4x - 1\).
• Combine like terms: the \(x\) terms \(2x\) and \(4x\) add up to \(6x\) and constants \(-6\) and \(-1\) sum to \(-7\).

After simplifying, the resulting expression, \(6x - 7\), reveals the linearity and characteristics of the new function, complete with its own slope and y-intercept, shedding light on how function operations alter algebraic and graphical properties.