## Mathematics for Economists

## Chapter 4 : Functions of one variable

### By Lund University

This chapter is devoted entirely to one of the most important concepts in mathematics namely functions. We begin the chapter by carefully looking at exactly what we mean by a function introducing the domain, codomain and range of a function. The most important class of functions are the linear functions which we will study extensively. Next, we look at some of the most important nonlinear functions. The chapter is concluded with a few slightly more advanced topics related to functions.

## Functions in general

In the first section we study the general concept of a function. A function is a rule that for each element in the domain assigns an element in the codomain. We then focus on real-valued functions of one real variable and look at graphs of such functions. Next, we investigate what happens to the graph of a function when we make changes, such as multiplying the function by a scalar. Finally we look at the definitions of increasing and decreasing functions and the strict versions of these concepts.

#### Introduction to functions

#### Natural domain

#### The graph of a function

#### The sum, product, difference and ratio of two functions

#### Shifting graphs

#### Increasing and decreasing functions

## Functions in general: Problems

Exercises on functions

#### Problem: Evaluate functions

#### Problem: Evaluate functions

#### Problem: Evaluate functions

#### Problem: Evaluate functions

#### Problem: Evaluate functions

#### Problem: Evaluate functions

#### Problem: Natural domain

#### Problem: Natural domain and range

#### Problem: Shifting graphs

## Linear functions

This section is entirely focused on the most important class of functions namely the linear functions. We will look at what we mean by the slope and the intercept of a linear function and we will look at how to determine the graph of a given linear function. We will also look at how to determine the equation of the function from, for example, two points on its graph.

#### Linear functions

#### Problem: Slope of a linear function

#### Problem: Graph of a linear function

#### Problem: Equations of linear functions from graphs

#### Problem: Equations of linear functions from two points or a point and a slope

#### Problem: System of equations and graphs of functions

#### Problem: System of equations with no solutions

#### Problem: Collections of points in the xy-plane

#### Problem: Collections of points in the xy-plane

#### Problem: Equations of a linear function from two points

#### Problem: Linear functions and the delta-notation

#### Problem: Equations of a linear function from a point and a slope

## Nonlinear functions

In this section we will look at the most common nonlinear functions. We begin with the quadratic functions and general polynomials and we study the graphs of these types of functions. Then we study the power functions, functions where the base in the power is our variable. For the exponential functions, it is the exponent in the power is our variable. This section is concluded with the logarithmic functions.

#### Quadratic functions and polynomials

#### Rational functions

#### Power functions

#### Exponential functions

#### Logarithmic functions

## Problems, Nonlinear functions

Exercises on nonlinear functions

#### Problem: Quadratic functions and vertex

#### Problem: Quadratic functions and vertex

#### Problem: Min and max of quadratic functions

#### Problem: Max of a quadratic function

#### Problem: Max of a quadratic function

#### Problem: Power function

#### Problem: Exponential function

#### Problem: Exponential function

#### Problem: Logarithmic function

#### Problem: Exponential function

#### Problem: Logarithmic function

#### Problem: Logarithm of a product

#### Problem: Log of a number close to 1

#### Problem: Logarithm of a power

#### Problem: Logarithm of an exponential function

#### Problem: Exponential and logarithmic function

#### Problem: Logarithm

#### Problem: Solving equations using log and exp

#### Problem: Logarithmic laws

## Composite and inverse functions

The final section of this chapter contains a few slightly more advanced topics. We begin with composite functions, where the basic idea is to put one function “inside” another function. We will learn how more complex functions can be written as a composition of simpler functions. We will then study properties of an injective function, a surjective function and a bijective function. From any bijective function we can create a new function called the inverse of the function. We will look at the problem of how to find the inverse of a bijective function. Finally, we will look at implicit relationships between two variables that in some cases may be converted into an explicit relationship or a function.

#### Composite functions

#### Injective, surjective and bijective functions

#### Inverse functions

#### Implicit relationship

## Inverse functions: Problems

Exercises on inverse functions