Functions 2.1 (A)

What is a function? Rene Descartes (1637) – Any positive integral power of a variable x. Gottfried Leibniz ( ) – Any quantity associated with a curve Leonhard Euler ( ) – Any equation with 2 variables and a constant Lejeune Dirichlet ( ) – Rule or correspondence between 2 sets

What is a relation? Step Brothers? Math Definition  Relation: A correspondence between 2 sets  If x and y are two elements in these sets, and if a relation exists between them, then x corresponds to y, or y depends on x  x  y or (x, y)

Example of relation NamesGrade on Ch. 1 Test Buddy A JimmyB KatieC Rob

Dodgeball Example Say you drop a water balloon off the top of a 64 ft. building. The distance (s) of the dodgeball from the ground after t seconds is given by the formula: Thus we say that the distance s is a function of the time t because:  There is a correspondence between the set of times and the set of distances  There is exactly one distance s obtained for any time t in the interval

Def. of a Function Let X and Y be two nonempty sets. A function from X into Y is a relation that associates with each element of X exactly one element of Y.  Domain: A pool of numbers there are to choose from to effectively input into your function (this is your x-axis).   The corresponding y in your function is your value (or image) of the function at x.   Range: The set of all images of the elements in the domain (This is your y-axis)

Domain/Range Example Determine whether each relation represents a function. If it is a function, state the domain and range. a) {(1, 4), (2, 5), (3, 6), (4, 7)} b) {1, 4), (2, 4), (3, 5), (6, 10)} c) {-3, 9), (-2, 4), (0, 0), (1, 1), (-3, 8)}

Practice Pg. 96 #2-12 Even

Function notation Given the equation Replace y with f(x)  f(x) means the value of f at the number x  x = independent variable  y = dependent variable

Finding values of a function For the function f defined by evaluate; a) f(3) b) f(x) + f(3) c) f(-x) d) –f(x) e) f(x + 3) f)

Practice 2 Pg. 96 #14, 18, 20

Implicit form of a function Implicit Form Explicit Form

Determine whether an equation is a function Is a function?

Finding the domain of a function Find the domain of each of the following functions:

Tricks to Domain Rule #1 If variable is in the denominator of function, then set entire denominator equal to zero and exclude your answer(s) from real numbers. Rule #2 If variable is inside a radical, then set the expression greater than or equal to zero and you have your domain!

Practice 3 Pg. 96 #22-46 E