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