1. Introduction to Calculus: Functions

2. Functions A function f is a rule that assigns an element f(x) of a set, called the target set of the function f, to any element x of the domain of definition of the function. The symbol x is called the variable of the function f. Definition Preliminaries/Introduction to Calculus/ Functions by M. Seppälä

3. Domain of Definition of a Function The set of values of the variable(s) of a function f for which the function is defined is called the domain of definition of the function f. Preliminaries/Introduction to Calculus/ Functions by M. Seppälä

4. A Thermometer is a Function This picture shows a reconstruction of a thermometer of Galileo. The sealed glass cylinder contains clear liquid and objects which sink differently as the cylinder is heated. The variable of the thermometer function is time, and the value of the function is the current temperature. Preliminaries/Introduction to Calculus/ Functions by M. Seppälä

5. Computer Programs are Functions function putDataXML(xmlDoc,sDocPath) { if(gnLoadFts==1) { var node=xmlDoc.lastChild; if(node) { var oChild=node.firstChild; var aFCD=new Array(); var aFTCD=new Array(); while(oChild) { if(oChild.nodeName=="chunkinfo") {..... A computer program produces an output of a given input, and is, therefore, a function. Even more, programs use subroutines that are functions themselves. Preliminaries/Introduction to Calculus/ Functions by M. Seppälä

6. Functions defined by Tables A function whose domain of definition is a finite set can also be defined by a table from which we can read the values of the function for all possible values of the variable. xf(x) aα bβ cγ dδ Preliminaries/Introduction to Calculus/ Functions by M. Seppälä

7. Functions in Calculus Functions in calculus are usually defined by algebraic expressions. The function f,, is defined for all x 1. Preliminaries/Introduction to Calculus/ Functions by M. Seppälä

8. Graphs of Functions A function f, whose domain of definition is a subset of the set of real numbers and whose values are real numbers, can be pictured by drawing the points (x, f(x)) for some values of x. x f(x) Preliminaries/Introduction to Calculus/ Functions by M. Seppälä

9. Notations of Functions The notation f: A B indicates that the domain of the definition of the function f is the set A, and that to each element of the set A the function f assigns an element of the set B. The set B is the target set of the function f. Preliminaries/Introduction to Calculus/ Functions by M. Seppälä

10. Functions It defines, therefore, a function, whose variable takes values in the interval [-1,1]. The values of this function are real numbers. The expression takes real values if -1 x 1. We indicate this by writing f: [-1,1]R,. Preliminaries/Introduction to Calculus/ Functions by M. Seppälä

11. Functions The function f: A B assigns, to any element a of the set A an element f(a) of the set B. The set A is the domain of definition of the function f, the set B is the target set of the function f. Preliminaries/Introduction to Calculus/ Functions by M. Seppälä The variable a of the function f varies in the set A. For any a A, f(a) B.

12. History of Functions The concept of a function was defined by Leonhard Euler in the mid 18th century. Using functions one can develop general theories. Preliminaries/Introduction to Calculus/ Functions by M. Seppälä

13. Leonhard Euler (1707 - 1783)