Functions 2.1 (A)

What is a function?

Rene Descartes (1637) – Any positive integral power of a variable x.

Gottfried Leibniz (1646-1716) – Any quantity associated with a curve

Leonhard Euler (1707-1783) – Any equation with 2 variables and a constant

Lejeune Dirichlet (1805-1859) – 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

Names Grade on Ch. 1 TestBuddy A

Jimmy BKatie CRob

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

€

s = 64 −16t 2

€

0 ≤ t ≤ 2

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 €

y = 2x −5

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1≤ x ≤ 6

Finding values of a function

For the function f defined byevaluate;

a) f(3)b) f(x) + f(3)c) f(-x)d) –f(x)e) f(x + 3)f)

€

f (x) = 2x 2 − 3x

€

f (x + h) − f (x)

h

Practice 2

Pg. 96 #14, 18, 20

Implicit form of a function

Implicit Form Explicit Form

€

3x + y = 5

x 2 − y = 6

xy = 4

€

y = f (x) = −3x +5

y = f (x) = x 2 −6

y = f (x) =4

x

Determine whether an equation is a function

Is a function?

€

x 2 + y 2 =1

Finding the domain of a function

Find the domain of each of the following functions:

€

f (x) = x 2 +5x

g(x) =3x

x 2 − 4

h(t) = 4 − 3t

Tricks to Domain

Rule #1If variable is in the denominator of function, then set entire denominator equal to zero and exclude your answer(s) from real numbers.

Rule #2If 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