# Functions

## Functions

A *function* is simply a rule that assigns a unique value of a dependent variable (e.g. \(f(x)\)) to each value of an independent variable (e.g. \(x\)): \(x \rightarrow f(x)\)

- Something is
*not*a function if it assigns*multiple*values of \(y\) for the same value of \(x\) (e.g. on a graph, a vertical line) - We can relate any
*independent*variable (e.g. \(x\)) to any*dependent*variable (e.g. \(y\))**so get comfortable using variables other than \(x\) and \(y\)!**

In its general form a function can be written as:

\[q = q(p)\]

- “Quantity (\(q\)) is a function of price (\(p\))”
- This expresses that there is a relationship between \(q\) and \(p\), it doesn’t tell us the
*specific*form of that relationship- \(q\) is the dependent or “endogenous” variable, its value is determined by \(p\)
- \(p\) is the independent or “exogenous” variable, its value is given and not dependent on other variables
- The specific form of this function might be:

\[q = 100-6p\]

- The numbers 100 and 6 are known as
**parameters**, they are parts of the quantitative relationship between quantity and price (the variables) that do not change - If we have values of \(p\), we can find the value of \(q(p)\):
- When \(p=10\) \[\begin{align*} q(p)&=100-6p\\ q(10)&=100-6(10)\\ q(10)&=100-60\\ q(10)&=40\\ \end{align*}\]
- When \(p=5\): \[\begin{align*} q(p)&=100-6p\\ q(5)&=100-6(5)\\ q(5)&=100-30\\ q(5)&=70\\ \end{align*}\]

**Multivariate** functions have multiple independent variables, such as:

\[q=f(k,l)\]

“Output (\(q\)) is a function of both capital (\(k\)) and labor (\(l\))”

In economics, we often restrict the

*domain*and*range*of functions to positive real numbers, \(\mathbb{R}_+\), since prices and quantities are never negative in the real world*Domain*: the set of \(x\)-values*Range*: the set of \(y\)-values determined by the function

## Inverse Functions

- Many functions have a useful
*inverse*, where we switch the independent variable and dependent variable- For example, if we have the demand function:

\[q=100-6p\]

we may want find the **inverse demand function**, an equation where \(p\) is the dependent variable, rather than \(q\) (this is how we normally graph Supply and Demand functions!) - To do this, we need to solve the above equation for \(p\):

\[ \begin{align*} q&=100-6p &&\text{The original equation}\\ q+6p&=100 && \text{Add } 6p \text{ to both sides}\\ 6p&=100-q && \text{Subtract } q \text{ from both sides}\\ p&=\frac{100}{6}-\frac{1}{6}q && \text{Divide both sides by 6}\\ \end{align*} \]

## Functions with Fractions

- Many people are rusty on a few useful algebra rules we will need, one being how to deal with fractions in equations
- To get rid of a fraction, multiply both sides of the equation by the fraction’s
*reciprocal*(swap the numerator and denominator), which will yield just 1

\[\begin{align*} 100&=\frac{1}{4}x & & \text{The equation to be solved for} x\\ \frac{4}{1} \big(\frac{100}{1}\big) &=\frac{4}{1}\bigg(\frac{1}{4}x\bigg) && \text{Multiplying by the reciprocal of } \frac{1}{4} \text{, which is } \frac{4}{1} \\ \frac{400}{1}&=\frac{4}{4}x & & \text{Cross multiplying fractions}\\ 400 & = x & & \text{Simplifying}\\ \end{align*}\]

- Alternatively (if possible), re-imagining the fraction as a decimal may help:

\[\begin{align*} 100&=\frac{1}{4}x && \text{The original equation}\\ 100&=0.25x && \text{Converting to a decimal}\\ 400 & = x && \text{Dividing both sides by }0.25\\ \end{align*}\]

*Add*fractions by finding a common denominator:

\[\begin{align*} \frac{4}{3} &+ \frac{2}{5}\\ \bigg(\frac{4 \times 5}{3 \times 5}\bigg) &+ \bigg(\frac{2 \times 3}{5 \times 3}\bigg)\\ \frac{20}{15}&+\frac{6}{15}\\ &= \frac{26}{15}\\ \end{align*}\]

*Multiply*fractions straight across the numerator and denominator

\[ \frac{4}{3} \times \frac{2}{5}=\frac{4 \times 2}{3 \times 5}=\frac{8}{15} \]