Download Functions

Functions
TS: Making decisions after reflection and review
Warm-Up:
Given f(x) = 2x – 3 what is f(-2)? f  2  2  2  3
f  2  4  3
f  2  7
What about 3f(-2)?
3  f  2
3  7
21
Objectives
To determine if a relation is a function.
 To use the vertical line test to decide
whether an equation defines a function.
 To find the domain and range of a
function.
 To use function notation to evaluate
functions.

Vocabulary
A relation is a set of
ordered pairs.
{ (1, 2), (2, 4), (3, 6) }
The domain of a
relation is the set of 1st
elements. (x’s)
{ 1, 2, 3 }
The range of a relation
is the set of 2nd
elements. (y’s)
{ 2, 4, 6 }
Functions
A function is a machine.
Functions
You put something in.
Functions
Information is processed.
Functions
You get something out.
Functions

A function is a relation in which every element in
the domain is paired with exactly one element in
the range.
PEOPLE
Michael
Tony
Yvonne
Justin
Dylan
Megan
Elizabeth
Emily
BLOOD
TYPE
A
B
AB
O
Function?
D
1
2
3
R
2
4
6
Function
D
7
8
9
R
4
5
Function
D
7
8
R
4
5
6
Not a
Function
Function?

{(1, 2), (3, 4), (5, 6), (7, 8)}
Function

{(1, 2), (3, 2), (5, 6), (7, 6)}
Function

{(1, 2), (1, 3), (5, 6), (5, 7)}
Not a Function
Vertical Line Test

If a vertical line can intersect a graph in
more than one point, then the graph is not
a function.
Function?
Function
Domain: (-, )
Not a Function
Domain: [-3, 3]
Range: [0, )
Range: [-3, 3]
Function?
Not a Function
Function
Domain: [0, )
Domain: (-, )
Range: (-, )
Range: (-, )
Function?
Not a Function
Function
Domain: (-, 0]
Domain: (-, )
Range: (-, )
Range: (-, 0]
Is Y a function of X?
x  y 1
x2  y 2  1
YES!
NO!
x  y 1
x  y 1
YES!
NO!
2
2
Functions
A function pairs one object with another.
 A function will produce only one object for
any pairing.
 A function can be represented by an
equation.

Functions
In order to distinguish one function from another we must name it.
Functions
Values that go into a function are independent.
Functions
Values that come out of a function are dependent.
Functions

To evaluate the function for a particular value,
substitute that value into the equation and
solve.

You can evaluate a function for an expression as
well as for a number.

To do so, substitute the entire expression into
the equation.

Be careful to include parentheses where needed.
Functions
yx
2
f  x   x2
the machine
f is a function of x that
produces x - squared
2
f  5  5  25
Find what y equals
when x equals 5.
Find f (5)
Function Notation
y  2x  3
f ( x)  2 x  3
Variable in the function
Name of the function
f (4)  2(4)  3
83
5
f ( x  3)  2( x  3)  3
 2x  6  3
 2x  9
Function Notation
h(t )  t 2  2t
h(3)  (3)  2(3)
2
96
3
h(t  2)  (t  2)  2(t  2)
2
 t 2  4t  4  2t  4
 t  2t
2
Piece-wise Function
 x 2  2, x  1
f ( x)  
 x  4, x  1
f (2)  (2) 2  2
6
f (0)  0  4
4
f (1)  (1)  2
3
f (3)  3  4
1
2
Function Notation
For f  x   x 2  4 x  7,
find f  x   f  x  x 
f  x  x    x  x  4  x  x  7
2
  x  x  x  x  4x 4x 7
 x  xx  xx   x  4x 4x 7
2
2
 x 2xx   x  4x 4x 7
2
2
The Difference Quotient
f  x  x   f  x 
For f  x   x  4 x  7, find
x
2
x  2 xx   x   4 x  4x  7   x  4 x  7 
2
2
2
x
2

x  2 xx   x   4 x  4x  7 x 4x 7
2
2
x
2 x x    x   4  x
x
2
The Difference Quotient
x  2 x  x  4 
x
2 x  x  4
Composition Functions
Given f(x) = 2x – 3 and g(x) = x2
f ( g ( x))  f g ( x)  2 g ( x)  3
 2( x )  3
2
 2x  3
2
but…
g ( f ( x))  g f ( x)  ( f ( x))
2
 (2 x  3)
2
 4 x  12 x  9
2
Conclusion

A function is a relationship between two sets
that pairs one object in the first set with one and
only one object in the second set.

To evaluate the function for a particular value,
substitute that value into the equation and
solve.

To evaluate the function for an expression,
substitute the entire expression into the
equation; include parentheses where needed.