Problem 42 - Calculus Early Transcendentals: Pearson New International Edition

Question

Which of the following functions satisfies \(f(x+y)=f(x)+f(y)\) for all real numbers \(x\) and \(y\) ? (a) \(f(t)=2 t\) (b) \(f(t)=t^{2}\) (c) \(f(t)=2 t+1\) (d) \(f(t)=-3 t\)

Step-by-Step Solution

Answer

Options (a) and (d) satisfy the equation.

Step 1: Understanding the functional equation

The equation given is \( f(x+y) = f(x) + f(y) \). This is a functional equation, and a function that satisfies this equation for all real numbers \(x\) and \(y\) is generally known as additive.

Step 2: Checking Option (a)

Consider \( f(t) = 2t \). Substitute into the functional equation: \( f(x+y) = 2(x+y) = 2x + 2y \). This equals \( f(x) + f(y) = 2x + 2y \). So, \( f(t) = 2t \) satisfies the equation.

Step 3: Checking Option (b)

Consider \( f(t) = t^2 \). Substitute into the functional equation: \( f(x+y) = (x+y)^2 \) and \( f(x) + f(y) = x^2 + y^2 \). \( (x+y)^2 = x^2 + 2xy + y^2\), which does not equal \( x^2 + y^2\). Therefore, \( f(t) = t^2 \) does not satisfy the equation.

Step 4: Checking Option (c)

Consider \( f(t) = 2t + 1 \). Substitute into the functional equation: \( f(x+y) = 2(x+y) + 1 = 2x + 2y + 1 \). The expected \( f(x) + f(y) = (2x + 1) + (2y + 1) = 2x + 2y + 2 \). Since \( 2x + 2y + 1 eq 2x + 2y + 2 \), it does not satisfy the equation.

Step 5: Checking Option (d)

Consider \( f(t) = -3t \). Substitute into the functional equation: \( f(x+y) = -3(x+y) = -3x - 3y \). This equals \( f(x) + f(y) = -3x - 3y \). So, \( f(t) = -3t \) satisfies the equation.

Step 6: Conclusion and selection

Both Option (a) \( f(t) = 2t \) and Option (d) \( f(t) = -3t \) satisfy the functional equation \( f(x+y) = f(x) + f(y) \). Therefore, options (a) and (d) are correct.

Key Concepts

Additive Functions

An additive function is a type of function that adheres to a simple yet pivotal property: for any inputs \( x \) and \( y \), the function satisfies the equation \( f(x+y) = f(x) + f(y) \). This characteristic makes them distinct from other functions and greatly simplifies many mathematical operations. Additive functions can be linear, where the function can be expressed as \( f(x) = cx \), where \( c \) is a constant, such as in options (a) \( f(t) = 2t \) and (d) \( f(t) = -3t \) from our exercise. They are pivotal in many areas of mathematics, especially in analysis and algebra, because they maintain structures and allow for predictable transformations of data. Notably, the additive property ensures that the change in output is directly proportional to the change in input.

Real Numbers

Real numbers are a fundamental mathematical concept that we use to describe quantities along the continuous line of numbers we often visualize numerically. They encompass rational numbers (those that can be expressed as a fraction, like 3 or 1/2) and irrational numbers (those that cannot be precisely written as a fraction, like \( \sqrt{2} \) or \( \pi \)). In the context of functional equations like \( f(x+y) = f(x) + f(y) \), real numbers serve as the broad domain over which these equations must hold true. This universality allows us to apply this principle to both everyday numbers and more complex ones without restriction. By definition, a function that can consistently satisfy these types of equations in the domain of real numbers is quite robust, as it must handle diverse and potentially complex values seamlessly.

Function Properties

Understanding the properties of a function is key to determining if it satisfies any given equation like \( f(x+y) = f(x) + f(y) \). Functions can have various properties such as linearity, continuity, and differentiability, which help categorize and manipulate them within mathematics. Additive functions often also demonstrate linearity, meaning they can be expressed in the form \( f(x) = ax + b \). This is evident from our analysis that functions like \( f(t) = 2t \) and \( f(t) = -3t \) exhibit linear behavior.These properties help in identifying the correct setup or type of function needed to solve problems, like setting up equations based on real-world data. They play a crucial role in formulating solutions to problems and understanding the behavior of dynamic systems in calculus and beyond.