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)
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)}
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
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
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!