Problem 54 - Calculus with Concepts in Calculus

Question

Let \(c\) be any number, and define a function \(f\) by \(f(x)=c x\). Show that \(f(x+y)=f(x)+f(y)\) for all numbers \(x\) and \(y\).

Step-by-Step Solution

Answer

The function \( f(x) = cx \) satisfies \( f(x+y) = f(x) + f(y) \) for all numbers \( x \) and \( y \).

Step 1: Express f(x+y)

The given function is \( f(x) = c \cdot x \). We begin by expressing \( f(x+y) \) using this function. Substitute \( x+y \) in place of \( x \) in the function definition: \[ f(x+y) = c(x+y) \]. This means we will multiply the constant \( c \) by the expression \( x+y \).

Step 2: Expand f(x+y) Using Distribution

To expand \( f(x+y) = c(x+y) \), apply the distributive property: \[ c(x+y) = c \cdot x + c \cdot y \]. This separates the multiplication of \( c \) with both \( x \) and \( y \).

Step 3: Relate Expanded Expression to f(x) and f(y)

Using the definitions of \( f(x) \) and \( f(y) \) from the function, we know \( f(x) = c \cdot x \) and \( f(y) = c \cdot y \). Substitute these into the expanded expression to get: \[ c \cdot x + c \cdot y = f(x) + f(y) \].

Step 4: Conclusion

By substituting back our expressions from the function \( f \), we verify \( f(x+y) = f(x) + f(y) \). This shows that the function \( f(x) = c \cdot x \) satisfies the property for all numbers \( x \) and \( y \).

Key Concepts

Linear Functions

Understanding linear functions can be much easier than you might think! A linear function is simply any function that can be represented by the equation of a straight line, which is usually expressed in the form \( f(x) = mx + b \). In the context of our exercise, the function \( f(x) = c \cdot x \) is a specific type of linear function where the line passes through the origin, because there is no constant term (meaning \( b = 0 \)).
Linear functions have several typical features that can help recognize them:

• They exhibit a constant rate of change, represented by the "slope" (in this case, the constant \( c \)).
• The graph is always a straight line.
• They follow a basic pattern that can be extended across various values.

The slope in our function is \( c \), which tells us how steep the line is. Simplifying the equation of a linear function, and understanding its graphical representation can provide valuable insights into how these functions behave in both pure mathematics and real-world applications.

Algebraic Proof

Algebraic proof is a powerful tool that allows us to validate truths about equations and expressions using algebraic techniques. In the solution to our exercise, the algebraic proof shows that \( f(x+y) = f(x) + f(y) \) by manipulating the expressions :
- First, we substituted \( x+y \) into the function, showing \( f(x+y) = c(x+y) \).- Then, by applying the distributive property, we expanded \( c(x+y) \) to \( c\cdot x + c\cdot y \).The next step involved recognizing that these expressions for \( c\cdot x \) and \( c\cdot y \) are exactly \( f(x) \) and \( f(y) \) respectively. This manipulation and substitution demonstrated the property desired in the exercise.
In algebraic proofs, using distributed properties, substitutions, and clear logical steps are crucial. It involves connecting concepts directly and showing that each step logically follows from the preceding one. This particular proof not only reinforces the function’s properties but also the inherent behaviours of linear functions.

Function Properties

Function properties are characteristics that define the behavior and rules governing a function. In our scenario, the property in focus is the addition property of a function, reflected in the equation \( f(x+y) = f(x) + f(y) \). This property shows that the function is additive.
This means any function satisfying \( f(x+y) = f(x) + f(y) \) is **linear** under certain conditions, especially when passing through the origin as is the case here. Some properties to recognize in linear functions include:

• Additivity: The combination of two inputs adds directly to yield the combination of their outputs.
• Homogeneity: If \( f(kx) = kf(x) \), where \( k \) is a constant, it strengthens its linear property.

Recognizing these properties can be particularly useful when identifying and working with linear functions. It can help predict outcomes, evaluate function combinations, and even solve complex problems by breaking them into simpler parts. Mastering these properties assists in both a deeper understanding of algebra and its practical applications.