# Inverse Demand Function

The inverse demand function is the demand function solved for price, meaning the price depends on the quantities. This function is used to graph the demand curve since the graph's axes are inverted. By convention, the dependent variable (quantity) is on the x-axis, and the independent variable (price) is on the y-axis.

## Direct Demand Function vs. Inverse Demand Function

The direct demand function is the demand function where the quantity demanded depends on the price.

The direct demand function is expressed as:

\[ Q_d = Q_d(p) \]

Where:

- \( Q_d \) represents the quantity demanded of a good or service.
- \( p \) is the price of the good or service.
- \( Q_d(p) \) is a function that shows how the quantity demanded \( Q_d \) varies in response to changes in the price \( p \).

An example of the direct demand function is:

$$ q = 800 - 10p $$

Now, starting from the direct demand function, we can solve for the price as a function of quantity as follows:

We start with the initial equation for the quantity demanded:

\[ q = 800 - 10p \]

Add \(10p\) to both sides of the equation:

\[ q + 10p = 800 \]

Subtract \(q\) from both sides to isolate the term with \(p\):

\[ 10p = 800 - q \]

Divide both sides of the equation by 10 to solve for \(p\):

\[ p = \frac{800 - q}{10} \]

Simplify the fraction:

\[ p = 80 - 0.1q \]

Note that in this function, the price depends on the quantity demanded. This is the function used to graph the demand curve:

This graph shows the quantity demanded depending solely on the price, with all other factors affecting demand, such as income and the prices of related goods (substitutes and complements), held constant or unchanged. With the inverse demand function, we can also find the direct demand function by solving for quantity in terms of price:

We start with the inverse demand function:

\[ p = 80 - 0.1q \]

Multiply both sides by 10 to eliminate the decimal coefficient:

\[ 10p = 10(80 - 0.1q) \]

Distribute the 10 on the right side:

\[ 10p = 800 - q \]

Add \(q\) to both sides to isolate \(q\) on one side of the equation:

\[ 10p + q = 800 \]

Subtract \(10p\) from both sides to obtain the direct demand function:

\[ q = 800 - 10p \]

In this way, it is possible to switch from the direct demand function to the inverse demand function and vice versa, depending on which function is needed.