Introduction to Business Variables

class: center, middle, inverse, title-slide

.title[
# Introduction to Business Variables
]
.subtitle[
## Session 01
]
.author[
### Alejandro Ucan
]
.date[
### 2023-09-17
]

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# Session's Goals

* Understand the concept of variable dependencies. 
 * Understand the concept of function and its difference with relations. 
 * Applications of functions in business.

---
# Motivation

__Q1:__ When you go to the gym, on what depends the amount of burned calories? 
--
 __Q2:__ When you are uploading a document, on what depends the time it takes to upload? 
--
 __Q3:__ When you pay taxes, on what depends the amount of taxes you pay? 
 
---
# Variable Dependency

> __Definition:__ Given two variables `$x$` and `$y,$` we say that `$y$` _depends_ on `$x$` if the value of `$y$` is determined by the value of `$x.$` This dependency implies a _relation_ between the possible values of `$x$` and `$y.$`

---
## Functions

> __Definition:__ A _function_ is a relation between two variables `$x$` and `$y$` such that for each value of `$x$` there is a unique value of `$y.$` We write `$y=f(x)$` to denote that `$y$` is a function of `$x.$`

---
### Parts of a function:
`$$f:D\to R$$` `$$x\mapsto y=f(x)$$`
* __Domain:__ the set of all possible values of `$x$` that can be inputs of the function. 
* __Codomain:__ the set of all possible values of `$y$` that can be outputs of the function. 
* __Relation Rule:__ the expression that relates `$x$` and `$y.$` 
---
## Example 1

> The price of a minute call in AT&T cost `$0.79$`mxn, then the cost of a call of `$x$` minutes is given by the function `$f(x)=0.79x.$` 
 * Domain: the set of all positive numbers. 
 * Codomain: the set of all positive real numbers. 
 * Relation rule: is given by the expression `$f(x)=0.79x.$`

---
## Example 2

> The profit of an enterprise depends on time, where its total revenue is `$1000t$` and its cost at time `$t$` is `$300t^2$`. 
* Domain: the set of all positive numbers. 
* Codomain: the set of all real numbers. 
* Relation rule: is given by the expression `$f(x)=1000t-300t^2.$`

---
## Graph of a function

> __Definition:__ the _graph_ of a function is a visual representation of the function `$f,$` where the domain is represented in the horizontal axis and the codomain in the vertical axis, and a point `$p$` is drawn at the coordinate `$(x,f(x)).$`

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## Example 3

> The graph of the function `$f(x)=0.79x$` is given by:

<div class="figure">
<img src="index_files/figure-html/unnamed-chunk-1-1.png" alt="Graph of the function `$f(x)=0.79x$`" width="100%" />
Graph of the function `$f(x)=0.79x$`
</div>
---
## Example 4

<div class="figure">
<img src="index_files/figure-html/unnamed-chunk-2-1.png" alt="Graph of the Temperature as time function" width="100%" />
Graph of the Temperature as time function
</div>
---
## The Supply function

> __Definition:__ the _supply function_ is a function that relates the price of a product with the quantity of the product that is supplied to the market. `$$p=f(q)$$` where `$p$` is the price of the product and `$q$` is the quantity of the product.

--
 
Notes on Supply function: 
* The supply function is an increasing function.

---
## The Demand function

> __Definition:__ the _demand function_ is a function that relates the price of a product with the quantity of the product that is demanded to the market. `$$p=f(q)$$` where `$p$` is the price of the product and `$q$` is the quantity of the product.

--
 
Notes on Demand function: 
* The demand function is a decreasing function.

---
### The equilibrium state

> __Definition:__ the _equilibrium state_ is the state where the supply and demand functions intersect.

<div class="figure">
<img src="index_files/figure-html/unnamed-chunk-3-1.png" alt="Example of an Equilibrium State" width="100%" />
Example of an Equilibrium State
</div>
---
## The Total Cost function

> __Definition:__ the _total cost function_ is a function that relates the quantity of a product with the total cost of producing that quantity. `$$C=f(q)$$` where `$C$` is the total cost and `$q$` is the quantity of the product.

---
## The Income function

> __Definition:__ the _income function_ represent the total amount of money obtained by selling a product. `$$I=f(q)$$` where `$I$` is the income and `$q$` is the quantity of the product.

---
## The Utility function

> __Definition:__ the _utility function_ is defined as the total income minus the cost functions. `$$U=f(q)$$` where `$U$` is the utility and `$q$` is the quantity of the product. `$$U=I-C$$`