1. Chapter 2
Functions
Zephaniah 3:17 The LORD your God is in your
midst, a mighty one who will save; he will rejoice
over you with gladness; he will quiet you by his
love; he will exult over you with loud singing.
3. 2.1 What is a function
A function is a relation where for any given input
there is exactly one output
4. 2.1 What is a function
A function is a relation where for any given input
there is exactly one output
A relation is a set of ordered pairs
5. 2.1 What is a function
A function is a relation where for any given input
there is exactly one output
A relation is a set of ordered pairs
Book Definition: A function f is a rule that
assigns to each element x in a set A exactly one
element, called f(x), to a set B
7. Input: x domain
Output: y range
Domain: the set of all “legal” inputs
Range: the set of all possible outputs
8. Input: x domain
Output: y range
Domain: the set of all “legal” inputs
Range: the set of all possible outputs
Notation
y = f(x)
9. Input: x domain
Output: y range
Domain: the set of all “legal” inputs
Range: the set of all possible outputs
Notation
y = f(x)
y depends on x
10. Input: x domain
Output: y range
Domain: the set of all “legal” inputs
Range: the set of all possible outputs
Notation
y = f(x)
y depends on x
x: independent variable
11. Input: x domain
Output: y range
Domain: the set of all “legal” inputs
Range: the set of all possible outputs
Notation
y = f(x)
y depends on x
x: independent variable
y: dependent variable
24. If f (x) = 2x − 3x + 4
2
find f (a + h)
2
f (a + h) = 2 ( a + h ) − 3( a + h ) + 4
25. If f (x) = 2x − 3x + 4
2
find f (a + h)
2
f (a + h) = 2 ( a + h ) − 3( a + h ) + 4
= 2 ( a + 2ah + h ) − 3a − 3h + 4
2 2
26. If f (x) = 2x − 3x + 4
2
find f (a + h)
2
f (a + h) = 2 ( a + h ) − 3( a + h ) + 4
= 2 ( a + 2ah + h ) − 3a − 3h + 4
2 2
2 2
= 2a + 4ah + 2h − 3a − 3h + 4
27. If f (x) = 2x − 3x + 4
2
find f (a + h)
2
f (a + h) = 2 ( a + h ) − 3( a + h ) + 4
= 2 ( a + 2ah + h ) − 3a − 3h + 4
2 2
2 2
= 2a + 4ah + 2h − 3a − 3h + 4
2 2
= 2a + 2h + 4ah − 3a − 3h + 4
28. A Piecewise Function is defined by different rules
on different parts of its domain.
29. A Piecewise Function is defined by different rules
on different parts of its domain.
⎧ 3 − x, x ≤ 3
f (x) = ⎨ 2
⎩ x , x > 3
30. A Piecewise Function is defined by different rules
on different parts of its domain.
⎧ 3 − x, x ≤ 3
f (x) = ⎨ 2 (review how to graph ... do on board)
⎩ x , x > 3
31. A Piecewise Function is defined by different rules
on different parts of its domain.
⎧ 3 − x, x ≤ 3
f (x) = ⎨ 2 (review how to graph ... do on board)
⎩ x , x > 3
Find f(2)
32. A Piecewise Function is defined by different rules
on different parts of its domain.
⎧ 3 − x, x ≤ 3
f (x) = ⎨ 2 (review how to graph ... do on board)
⎩ x , x > 3
Find f(2)
f (2) = 3 − 2
=1
33. A Piecewise Function is defined by different rules
on different parts of its domain.
⎧ 3 − x, x ≤ 3
f (x) = ⎨ 2 (review how to graph ... do on board)
⎩ x , x > 3
Find f(2)
f (2) = 3 − 2
=1
Find f(4)
34. A Piecewise Function is defined by different rules
on different parts of its domain.
⎧ 3 − x, x ≤ 3
f (x) = ⎨ 2 (review how to graph ... do on board)
⎩ x , x > 3
Find f(2)
f (2) = 3 − 2
=1
Find f(4)
f (4) = 4 2
= 16
44. Finding Domains ...
State the Domain for each function:
4
f (x) = 2
x +x
2
x +x≠0
45. Finding Domains ...
State the Domain for each function:
4
f (x) = 2
x +x
2
x +x≠0
x ( x + 1) ≠ 0
46. Finding Domains ...
State the Domain for each function:
4
f (x) = 2
x +x
2
x +x≠0
x ( x + 1) ≠ 0
x ≠ 0 and x + 1 ≠ 0
47. Finding Domains ...
State the Domain for each function:
4
f (x) = 2
x +x
2
x +x≠0
x ( x + 1) ≠ 0
x ≠ 0 and x + 1 ≠ 0
{x : x ≠ −1,0}
48. Finding Domains ...
State the Domain for each function:
4
f (x) = 2 f (x) = 16 − x 2
x +x
2
x +x≠0
x ( x + 1) ≠ 0
x ≠ 0 and x + 1 ≠ 0
{x : x ≠ −1,0}
49. Finding Domains ...
State the Domain for each function:
4
f (x) = 2 f (x) = 16 − x 2
x +x
2 2
x +x≠0 16 − x ≥ 0
x ( x + 1) ≠ 0
x ≠ 0 and x + 1 ≠ 0
{x : x ≠ −1,0}
50. Finding Domains ...
State the Domain for each function:
4
f (x) = 2 f (x) = 16 − x 2
x +x
2 2
x +x≠0 16 − x ≥ 0
x ( x + 1) ≠ 0 2
x ≤ 16
x ≠ 0 and x + 1 ≠ 0
{x : x ≠ −1,0}
51. Finding Domains ...
State the Domain for each function:
4
f (x) = 2 f (x) = 16 − x 2
x +x
2 2
x +x≠0 16 − x ≥ 0
x ( x + 1) ≠ 0 2
x ≤ 16
x ≠ 0 and x + 1 ≠ 0 x ≥ −4 and x ≤ 4
{x : x ≠ −1,0}
52. Finding Domains ...
State the Domain for each function:
4
f (x) = 2 f (x) = 16 − x 2
x +x
2 2
x +x≠0 16 − x ≥ 0
x ( x + 1) ≠ 0 2
x ≤ 16
x ≠ 0 and x + 1 ≠ 0 x ≥ −4 and x ≤ 4
{x : x ≠ −1,0}
53. Finding Domains ...
State the Domain for each function:
4
f (x) = 2 f (x) = 16 − x 2
x +x
2 2
x +x≠0 16 − x ≥ 0
x ( x + 1) ≠ 0 2
x ≤ 16
x ≠ 0 and x + 1 ≠ 0 x ≥ −4 and x ≤ 4
{x : x ≠ −1,0}
{x : −4 ≤ x ≤ 4}
54. Finding Domains ...
State the Domain for each function:
4
f (x) = 2 f (x) = 16 − x 2
x +x
2 2
x +x≠0 16 − x ≥ 0
x ( x + 1) ≠ 0 2
x ≤ 16
x ≠ 0 and x + 1 ≠ 0 x ≥ −4 and x ≤ 4
{x : x ≠ −1,0}
{x : −4 ≤ x ≤ 4}
[ −4, 4 ]
56. Finding Domains ...
State the Domain for each function:
2
x +x−2
f (x) = 2
x + 5x + 6
57. Finding Domains ...
State the Domain for each function:
2
x +x−2
f (x) = 2
x + 5x + 6
f (x) =
( x + 2 )( x − 1)
( x + 2 )( x + 3)
58. Finding Domains ...
State the Domain for each function:
2
x +x−2
f (x) = 2
x + 5x + 6
f (x) =
( x + 2 )( x − 1)
( x + 2 )( x + 3)
{x : x ≠ −3,−2}
59. Bonus Math ...
Let’s look at the graph for that function:
2
x +x−2
f (x) = 2
x + 5x + 6
60. Bonus Math ...
Let’s look at the graph for that function:
2
x +x−2
f (x) = 2
x + 5x + 6
x −1
Behaves like: f (x) =
x+3
61. Bonus Math ...
Let’s look at the graph for that function:
2
x +x−2
f (x) = 2
x + 5x + 6
x −1
Behaves like: f (x) =
x+3
vertical asymptote at x=-3
62. Bonus Math ...
Let’s look at the graph for that function:
2
x +x−2
f (x) = 2
x + 5x + 6
x −1
Behaves like: f (x) =
x+3
vertical asymptote at x=-3
hole at x=-2
63. Bonus Math ...
Let’s look at the graph for that function:
2
x +x−2
f (x) = 2
x + 5x + 6
x −1
Behaves like: f (x) =
x+3
vertical asymptote at x=-3
hole at x=-2
(graph both to verify)
64. Do HW #1
“Always do more than what is required of you.”
George S. Patton
