Problem 63 - Calculus an Applied Approach

Question

Demand Function Given \(f(x)=x+1,\) which function would most likely represent a demand function? Explain your reasoning. Use a graphing utility to graph each function, and use each graph as part of your explanation. (a) \(p=f(x) \quad\) (b) \(p=x f(x) \quad\) (c) \(p=-f(x)+5\)

Step-by-Step Solution

Answer

Out of the three options, \( p = -f(x) + 5 \) is the function that represents the demand function as it possesses the characteristic properties of a demand function: a negative slope implying that as the price decreases, quantity demanded increases.

Step 1: Understand the nature of demand function

A demand function typically shows how quantity demanded (x) relates to price (p). It often has negative slope, representing the law of demand: as price increases, quantity demanded decreases.

Step 2: Graph and examine \( p = f(x) \)

By replacing \( f(x) \) with \( x+1 \), this function becomes \( p = x+1 \). It's straight line with a positive slope. Since demand functions typically have negative slope, this is unlikely to be a demand function.

Step 3: Graph and examine \( p = xf(x) \)

By replacing \( f(x) \) with \( x+1 \), this function becomes \( p = x(x+1) = x^2 + x \). This is a quadratic function opening upward. It doesn't represent the law of demand and hence this is also not likely a demand function.

Step 4: Graph and examine \( p = -f(x) + 5 \)

By replacing \( f(x) \) with \( x+1 \), this function becomes \( p = -(x+1) + 5 = -x+4 \). This is straight line with a negative slope. It accurately represents the law of demand that as price (p) decreases, quantity (x) increases. Hence, \( p = -f(x) + 5 \) represents the demand function.

Key Concepts

Law of Demand

The concept of the "law of demand" is fundamental in economics. It describes how the quantity demanded of a good or service changes in response to its price. Simply put, as the price of a good increases, the quantity demanded typically decreases, and vice versa. This relationship is visualized as a downward-sloping demand curve.

When you plot a demand function on a graph, the law of demand suggests that the curve should naturally slope from the top left to the bottom right. This negative slope indicates the inverse relationship between price and quantity demanded.

• Higher price leads to lower demand.
• Lower price encourages higher demand.

Understanding this principle helps in analyzing consumer behavior and predicting how shifts in pricing might affect demand.

Quadratic Function

Quadratic functions are a type of polynomial function characterized by the presence of an \[x^2\] term. The general form is \(y = ax^2 + bx + c\). Unlike linear functions, which are represented as straight lines, quadratic functions create parabolic curves when graphed.

In the context of demand functions, quadratic functions can sometimes appear, but they typically don't align with the law of demand if they open upwards. The curve of a quadratic function can either open upwards or downwards:

• Upward opening: The graph is U-shaped.
• Downward opening: The graph is n-shaped.

The function \(p = x^2 + x\) opens upward, depicting an increase in price with even larger increases as \(x\) grows, which does not fit the typical demand pattern of decreasing quantity with increasing price.

Negative Slope

The "negative slope" is crucial when identifying valid demand functions. A negative slope in a demand function graph reinforces the law of demand, visually confirming the inverse relationship between price and quantity.
Consider a function like \(p = -x + 4\): the slope is negative. As you move along the graph, each step to the right—indicating an increase in quantity—correlates with a decrease in price.

Such graphs logically model the expected consumer response to pricing, which aligns with the actual behavior observed in markets. This property is important because it depicts:

• Price decreases as quantity increases.
• Price increases as quantity decreases.

Thus, negative slopes are a signifier of demand functions that abide by the principles of economic supply-and-demand mechanics.