Functions 2.1 (A)
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Transcript Functions 2.1 (A)
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
Buddy
Jimmy
Katie
Rob
Grade on Ch. 1 Test
A
B
C
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:
s 64 16t
2
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 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
y 2x 5
1 x 6
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) f (x h) f (x)
h
f (x) 2x 3x
2
Practice 2
Pg. 96 #14, 18, 20
Implicit form of a function
Implicit Form
Explicit Form
3x y 5
y f (x) 3x 5
x y 6
xy 4
y f (x) x 6
2
2
4
y f (x)
x
Determine whether an equation is a function
Is
x 2 y 2 1 a function?
Finding the domain of a function
Find the domain of each of the following functions:
f (x) x 5x
2
3x
g(x) 2
x 4
h(t) 4 3t
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