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