Problem 121 - College Algebra with Modeling and Visualization

Question

Applying Concepts Show that the sum of two linear functions is a linear function.

Step-by-Step Solution

Answer

The sum of two linear functions is a linear function because it forms another expression of the same linear form.

Step 1: Define the Linear Functions

Start by defining two linear functions. Let function \( f(x) \) be defined as \( f(x) = ax + b \) and function \( g(x) \) be defined as \( g(x) = cx + d \), where \(a\), \(b\), \(c\), and \(d\) are constants.

Step 2: Add the Functions

The sum of the two functions, \( f(x) + g(x) \), can be computed by adding the expressions of \( f(x) \) and \( g(x) \). This means evaluating: \[ f(x) + g(x) = (ax + b) + (cx + d) \].

Step 3: Simplify the Expression

Combine the like terms of the sum \((ax + cx)\) and \((b + d)\). This results in: \[ f(x) + g(x) = (a + c)x + (b + d) \].

Step 4: Identify the Resulting Function

Observe that the simplified expression \((a + c)x + (b + d)\) is of the form \( mx + n \), where \( m = a+c \) and \( n = b+d \). This is the general form of a linear function.

Step 5: Conclude with Justification

Since the expression \((a + c)x + (b + d)\) adheres to the form \( mx + n \), it is confirmed that the sum of the two linear functions is itself a linear function.

Key Concepts

Function Addition

Function addition is a way to combine two or more functions to create a new function. When you add two functions, you're essentially adding their outputs together for every input value.

Here's how it relates to linear functions:

• Linear functions typically take the form \( f(x) = ax + b \).
• To add two linear functions like \( f(x) = ax + b \) and \( g(x) = cx + d \), you add them as \( f(x) + g(x) = (ax + b) + (cx + d) \).
• The result is \( (a + c)x + (b + d) \), which is another linear equation.

This shows that no matter how many linear functions you add together, the result will still be a linear function itself.

The beauty of function addition lies in its simplicity, and it's a fundamental concept in algebra and calculus alike.

Algebraic Expressions

Algebraic expressions are mathematical phrases that involve using numbers, variables, and arithmetic operations. They are the building blocks of equations and inequalities and help us express mathematical relationships.

• The expression \( ax + b \) represents a basic linear expression, where \( a \) is the coefficient of the variable \( x \), and \( b \) is the constant term.
• In our exercise, we added \( (ax + b) \) and \( (cx + d) \).
• To simplify them, we merged like terms to create the new expression \( (a + c)x + (b + d) \).

Understanding how to manipulate algebraic expressions is crucial. It helps in simplifying complex problems and is foundational in higher-level math and various applications in science and engineering. Algebra gives us a universal language to formulate real-world situations into solvable mathematical problems.

Mathematical Proofs

Mathematical proofs are logical arguments that establish the truth of mathematical statements. They are essential for validating concepts and theories in mathematics.

The process of mathematical proofs typically involves:

• Start with known information or assumptions.
• Using logical steps, show that a conclusion must follow.

In our linear function addition exercise, we used a proof to show that the sum of two linear functions is always linear.

Here's how the proof worked:

• We started with two known linear functions \( f(x) = ax + b \) and \( g(x) = cx + d \).
• We added them together to get \( (a + c)x + (b + d) \).
• The conclusion was that this sum \( (a + c)x + (b + d) \) is also a linear function because it fits the structure \( mx + n \).

The logic used in mathematical proofs is powerful. It confirms results and builds new mathematics on a solid foundation.