Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Abuse of notation wikipedia , lookup
Functional decomposition wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Big O notation wikipedia , lookup
Principia Mathematica wikipedia , lookup
Continuous function wikipedia , lookup
Dirac delta function wikipedia , lookup
Non-standard calculus wikipedia , lookup
Elementary mathematics wikipedia , lookup
Multiple integral wikipedia , lookup
History of the function concept wikipedia , lookup
SECTION 1.1 FUNCTIONS DEFINITION OF A RELATION A relation is a correspondence between two sets. If x and y are two elements in these sets and if a relation exists between x and y, we say that x corresponds to y or that y depends upon x. The correspondence can be written as an ordered pair (x,y). DEFINITION OF A RELATION Thus, a relation is simply a set of ordered pairs or a table which relates x and y values. DEFINITION OF FUNCTION Let X and Y be two nonempty sets of real numbers. A function from X into Y is a rule or a correspondence that associates with each element of X a unique element of Y. This is a special type of relation. For every x, there is only one y! DOMAIN AND RANGE DEFINITION The set of all x values. OF DOMAIN DEFINITION The set of all y values. OF Also called “functional RANGE values”. THE FUNCTION AS A “MAPPING” x-values y-values 1 8 4 2 7 0 -3 -2 DOMAIN RANGE Ordered Pairs: (1 , 2) (4 , 8) (7, - 3) (- 2, 0) THE FUNCTION AS A “MAPPING” Consider 3 students whose names are mapped to their letter grades on the last History exam: Jill A Frank B Sue C For each person in the domain, there can be only one associated letter grade in the range. THE SQUARING FUNCTION -2 -1 0 0 1 1 4 2 3 9 Each element in the domain maps to its square. COUNTER-EXAMPLE: 4 1 2 5 3 Ordered Pairs: (4, 1) (4, 2) (5, 3) This is an example of a relation but not a function. THREE WAYS TO REPRESENT A FUNCTION NUMERICALLY - ordered pairs SYMBOLICALLY - equation GRAPHICALLY - picture EXAMPLE Determine whether the relation represents a function: (a) {(1,4),(2,5),(3,6),(4,7)} EXAMPLE Determine whether the relation represents a function: (a) {(1,4),(2,4),(3,5),(6,10)} EXAMPLE Determine whether the relation represents a function: (a) {( - 3,9),(- 2,4),(0,0),(1,1),( - 3,8)} EXAMPLE Determine whether the relation represents a function: (a) {( - 3,9),(- 2,4),(0,0),(1,1),( - 3,8)} EVALUATING A FUNCTION AT A GIVEN X-VALUE x f(x) f(0) = 0 0 0 4 2 4 f(-2) = 4 -2 4 f(9) = 81 9 81 f(x) = x f(2) = 2 Symbolically, the squaring function can be represented as y = x2 “FUNCTIONAL NOTATION” f(x) = x 2 Read: “f of x equals x squared” EVALUATING A FUNCTION AT A GIVEN X-VALUE For f(x) = 2x2 – 3x, find the values of the following: (a) f(3) (b) f(x) + f(3) (d) - f(x) (e) f(x + 3) (f) f(x h) f(x) h (c) f(-x) FINDING VALUES OF A FUNCTION ON A CALCULATOR DO EXAMPLE 7 IMPLICIT FORM OF A FUNCTION When a function is defined by an equation in x and y, we say that the function is given implicitly. If it is possible to solve the equation for y in terms of x, then we write y = f(x) and say that the function is given explicitly. See examples on Pg 127. DETERMINING WHETHER AN EQUATION IS A FUNCTION Determine if x2 + y2 = 1 is a function. y 1 x 2 This means that for certain values of x, there are two possible outcomes for y. Thus, this is not a function! Important Facts About Functions: 1. For each x in the domain of a function f, there is one and only one image f(x) in the range. For every x, there is only one y. 2. f is the symbol we use to denote the function. It is symbolic of the equation that we use to get from an x in the domain to f(x) in the range. f(x) is another name for y. Important Facts About Functions: 3. If y = f(x) , then x is called the independent variable or argument of f, and y is called the dependent variable or the value of f at x. DOMAIN OF A FUNCTION If a function is being described symbolically and it comes with a specific domain, that domain should be expressly given. Otherwise, the domain of the function will be assumed to be the “natural domain”. EXAMPLE: f(x) = 2 x Knowing the function of squaring a number, we can determine that the natural domain is all real numbers because any real number can be squared. We can also look at a graph. EXAMPLE: f(x) = 2 x + 5x This is simply a modification of the squaring function. Thus, we can determine that the natural domain is all real numbers. We can also look at a graph. EXAMPLE Find the domain: x 2 D: { x x 2 } 3x f(x) = 2 x 4 EXAMPLE Find the domain: 4 – 3t 0 3t 4 t 3 4 f(t) = 4 - 3t EXAMPLE Find the domain : f(x) = x3 + x - 1 All real numbers OPERATIONS ON FUNCTIONS Notation for four basic operations on functions: (f + g)(x) = f(x) + g(x) (f - g)(x) = f(x) - g(x) (f g)(x) = f(x) g(x) (f / g)(x) = f(x) / g(x) OPERATIONS ON FUNCTIONS Do Example 10 CONCLUSION OF SECTION 1.1