Problem 123 - Precalculus : Building Concepts and Connections

Question

Let \(f(x)=a x+b\) and \(g(x)=c x+d,\) where \(a, b, c,\) and \(d\) are constants. Show that \((f+g)(x)\) and \((f-g)(x)\) also represent linear functions.

Step-by-Step Solution

Answer

Both \((f+g)(x)\) and \((f-g)(x)\) represent linear functions as their forms are still of the form \(mx + n\), which is the form of a linear function.

Step 1: The Sum of Functions

Given two functions, \(f(x)\) and \(g(x)\), the sum can be obtained by directly adding these two functions. Therefore, \((f+g)(x) = f(x) + g(x) = (ax + b) + (cx + d) = (a+c)x + (b+d)\). We can observe that \((f+g)(x)\) is in the form of a linear function as \(m=a+c\) and \(n=b+d\). So, the sum of two given linear functions represents a linear function.

Step 2: The Difference of Functions

In the same manner, the difference of two functions can be calculated by subtracting \(g(x)\) from \(f(x)\). Therefore, \((f-g)(x) = f(x) - g(x) = (ax + b) - (cx + d) = (a-c)x + (b-d)\). Here again, \((f-g)(x)\) is in the form of a linear function as \(m=a-c\) and \(n=b-d\). Hence, the difference of the two given linear functions also represents a linear function.

Key Concepts

Function Addition

When adding two linear functions, you are simply combining their respective coefficients and constants. For example, consider two linear functions: \( f(x) = ax + b \) and \( g(x) = cx + d \).

• To find their sum, you compute \( (f+g)(x) = f(x) + g(x) \).
• This translates to \( (ax + b) + (cx + d) \).

By rearranging, you achieve the expression \( (a+c)x + (b+d) \).
Notice that the result remains a linear function because it holds the form \( mx + n \), where \( m \) is the combined slope \( (a+c) \) and \( n \) is the combined y-intercept \( (b+d) \).
This shows how straightforward linear function addition is. Just focus on adding corresponding parts! No complicated transformations involved.

Function Subtraction

Just like with addition, subtraction of two linear functions involves handling their individual parts. Let's take the same functions: \( f(x) = ax + b \) and \( g(x) = cx + d \).

• Function subtraction is expressed as \( (f-g)(x) = f(x) - g(x) \).
• Here, you calculate \( (ax + b) - (cx + d) \).

This simplifies to \( (a-c)x + (b-d) \), which is once again a linear function.
The slope \( a-c \) and y-intercept \( b-d \) continue to preserve the linear nature.
Subtraction mirrors addition, just remember to subtract the coefficients and constants accordingly.

Algebraic Expressions

Algebraic expressions, such as those we encounter in linear functions, are expressions made up of variables, constants, and arithmetic operations. In our context, the functions \( f(x) = ax + b \) and \( g(x) = cx + d \) are algebraic expressions that define straight lines.

• They are called linear because their highest power of \( x \) is one.
• Each term in the form of \( mx \) indicates a change rate (slope), while constant \( n \) represents the y-intercept.

Understanding the basic structure of algebraic expressions is key.
These expressions involve operations like addition and subtraction, leading to new expressions such as \( (f+g)(x) \) and \( (f-g)(x) \).
Algebra helps us simplify, manipulate, and interpret these functions in valuable ways.