Problem 48 - Calculus A New Horizon

Question

In each part, find functions \(f\) and \(g\) that are increasing on \((-\infty,+\infty)\) and for which \(f-g\) has the stated property. (a) \(f-g\) is decreasing on \((-\infty,+\infty)\) (b) \(f-g\) is constant on \((-\infty,+\infty)\) (c) \(f-g\) is increasing on \((-\infty,+\infty)\)

Step-by-Step Solution

Answer

(a) \( f(x) = x, g(x) = 2x \); (b) \( f(x) = x, g(x) = x+c \); (c) \( f(x) = 2x, g(x) = x \).

Step 1: Understanding the Problem

We are tasked with finding two functions, \( f \) and \( g \), both increasing on the entire real line, such that the difference \( f-g \) meets specific criteria for being decreasing, constant, or increasing.

Step 2: Finding Functions for Part (a)

For part (a), we need both \( f \) and \( g \) to be increasing, but their difference \( f-g \) must be decreasing. One way to achieve this is to have \( f(x) = x \) and \( g(x) = 2x \). Here, both functions are increasing since their derivatives are positive. However, the derivative of \( f-g = x - 2x = -x \) is \( -1 \), which is negative, making the difference function decreasing.

Step 3: Finding Functions for Part (b)

For part (b), \( f-g \) needs to be constant. We can choose \( f(x) = x \) and \( g(x) = x + c \) where \( c \) is a constant. Both functions are increasing due to their positive derivatives. The difference \( f-g = x - (x+c) = -c \) is a constant function, meeting the given condition.

Step 4: Finding Functions for Part (c)

In part (c), we want \( f-g \) to be increasing. We can select \( f(x) = 2x \) and \( g(x) = x \). Both functions are increasing, and now the difference \( f-g = 2x - x = x \) has a derivative of 1, which is positive, thus making it an increasing function.

Key Concepts

Function Differences

Function differences involve subtracting one function from another. The task is to analyze how the resulting function behaves. In the context of this exercise, we deal with increasing functions, but we are particularly interested in how the difference between two functions, \( f - g \), behaves as we move along the real line.

• If \( f-g \) is decreasing, it means that the rate of increase of \( g \) surpasses the rate of increase of \( f \).


• If \( f-g \) is constant, both \( f \) and \( g \) increase at the same rate.


• When \( f-g \) is increasing, \( f \) increases at a faster rate than \( g \).

Understanding function differences allows us to predict and describe the overall behavior of the composite function.

Real Line

The real line, denoted by \((-\infty,+\infty)\), is the set of all real numbers. It is an important concept as it represents a continuum of values without any gaps.

• In this exercise, both functions \( f \) and \( g \) are examined over the entire real line, meaning they apply universally to all real number inputs.


• The behavior of the difference \( f-g \) is also described over this infinite domain.

Working with functions on the real line helps us understand their properties globally, providing us with comprehensive insights into their behaviors across all possible inputs.

Derivatives

Derivatives represent the rate of change of a function with respect to changes in its input. They help determine whether a function is increasing or decreasing.

• If a function's derivative is positive, the function is increasing.


• If its derivative is negative, the function is decreasing.

In the context of the exercise, we ensure that both functions \( f \) and \( g \) are increasing by choosing functions whose derivatives are positive:

• Example: For \( f(x) = x \), the derivative is \( f'(x) = 1 \).


• For \( g(x) = 2x \), the derivative is \( g'(x) = 2 \).

The derivative of \( f - g \) determines whether this difference remains constant, increases, or decreases. Thus, derivatives are pivotal in analyzing function differences and their behavior.

Constant Function

A constant function is a function that does not change; its output remains the same regardless of the input. It can be easily identified as a flat line on a graph.

• For example, \( h(x) = c \), where \( c \) is a constant, is a constant function.

In this exercise, we explore a scenario where the difference \( f - g \) is constant. This occurs when the rates at which the functions \( f \) and \( g \) increase are identical.

• By choosing \( f(x) = x \) and \( g(x) = x + c \), their difference \( f-g = -c \) is constant if \( c \) is a fixed number.

The concept of constant functions is crucial as it tells us about a uniform output, regardless of input variations, especially in differentiating increasing or decreasing functions.