1. The Basic Language of Functions

2. The Basic Language of Functions
A function is a procedure that assigns each input
exactly one output.

3. The Basic Language of Functions
A function is a procedure that assigns each input
exactly one output. In mathematics, usually we name
functions as f, g, h…and we let x represent the input
and y represent the output.

4. The Basic Language of Functions
A function is a procedure that assigns each input
exactly one output. In mathematics, usually we name
functions as f, g, h…and we let x represent the input
and y represent the output.
Example A. a. Input a number x, the output is the
largest integer which is not more than x

5. The Basic Language of Functions
A function is a procedure that assigns each input
exactly one output. In mathematics, usually we name
functions as f, g, h…and we let x represent the input
and y represent the output.
Example A. a. Input a number x, the output is the
largest integer which is not more than x (e.g. if x = 1/2,
the output is y = 0),

6. The Basic Language of Functions
A function is a procedure that assigns each input
exactly one output. In mathematics, usually we name
functions as f, g, h…and we let x represent the input
and y represent the output.
Example A. a. Input a number x, the output is the
largest integer which is not more than x (e.g. if x = 1/2,
the output is y = 0), is this procedure a function?
1/2
–3 –2 –1 0 1 2

7. The Basic Language of Functions
A function is a procedure that assigns each input
exactly one output. In mathematics, usually we name
functions as f, g, h…and we let x represent the input
and y represent the output.
Example A. a. Input a number x, the output is the
largest integer which is not more than x (e.g. if x = 1/2,
the output is y = 0), is this procedure a function?
1/2
–3 –2 –1 0 1 2
The largest integer not more than 1/2

8. The Basic Language of Functions
A function is a procedure that assigns each input
exactly one output. In mathematics, usually we name
functions as f, g, h…and we let x represent the input
and y represent the output.
Example A. a. Input a number x, the output is the
largest integer which is not more than x (e.g. if x = 1/2,
the output is y = 0), is this procedure a function?
1/2
–3 –2 –1 0 1 2
The largest integer not more than 1/2
This is a function because for each x, there is exactly
one "largest integer" which is less than or equal to x.

9. The Basic Language of Functions
A function is a procedure that assigns each input
exactly one output. In mathematics, usually we name
functions as f, g, h…and we let x represent the input
and y represent the output.
Example A. a. Input a number x, the output is the
largest integer which is not more than x (e.g. if x = 1/2,
the output is y = 0), is this procedure a function?
1/2
–3 –2 –1 0 1 2
The largest integer not more than 1/2
This is a function because for each x, there is exactly
one "largest integer" which is less than or equal to x.
This function is known as the greatest integer or the
truncation function which is denoted as [x].

10. The Basic Language of Functions
A function is a procedure that assigns each input
exactly one output. In mathematics, usually we name
functions as f, g, h…and we let x represent the input
and y represent the output.
Example A. a. Input a number x, the output is the
largest integer which is not more than x (e.g. if x = 1/2,
the output is y = 0), is this procedure a function?
1/2
–3 –2 –1 0 1 2
The largest integer not more than 1/2
This is a function because for each x, there is exactly
one "largest integer" which is less than or equal to x.
This function is known as the greatest integer or the
truncation function which is denoted as [x].
Hence [1/2] = [0.277] = [0] = 0.

11. The Basic Language of Functions
b. Input a number x, the output is (are) integer(s)
within ¾ of x. Is this a function?

12. The Basic Language of Functions
b. Input a number x, the output is (are) integer(s)
within ¾ of x. Is this a function?
No, this is not a function because if x = ½,
1/2
–1 0 1 2

13. The Basic Language of Functions
b. Input a number x, the output is (are) integer(s)
within ¾ of x. Is this a function?
No, this is not a function because if x = ½,
¾ ¾
1/2
–1 0 1 2

14. The Basic Language of Functions
b. Input a number x, the output is (are) integer(s)
within ¾ of x. Is this a function?
No, this is not a function because if x = ½, there'll be
two different outputs y = 0 or 1.
¾ ¾
1/2
–1 0 1 2
The two integers within ¾ of 1/2

15. The Basic Language of Functions
b. Input a number x, the output is (are) integer(s)
within ¾ of x. Is this a function?
No, this is not a function because if x = ½, there'll be
two different outputs y = 0 or 1.
¾ ¾
1/2
–1 0 1 2
The two integers within ¾ of 1/2
For a procedure to be a function, it must produce a
unique, i.e. one and only, output for every input x.

16. The Basic Language of Functions
b. Input a number x, the output is (are) integer(s)
within ¾ of x. Is this a function?
No, this is not a function because if x = ½, there'll be
two different outputs y = 0 or 1.
¾ ¾
1/2
–1 0 1 2
The two integers within ¾ of 1/2
For a procedure to be a function, it must produce a
unique, i.e. one and only, output for every input x.
Note that in the above example if x = 2.1, then 2 is
only one integer that’s with ¾ of 2.1, i.e. y = 2.

17. The Basic Language of Functions
b. Input a number x, the output is (are) integer(s)
within ¾ of x. Is this a function?
No, this is not a function because if x = ½, there'll be
two different outputs y = 0 or 1.
¾ ¾
1/2
–1 0 1 2
The two integers within ¾ of 1/2
For a procedure to be a function, it must produce a
unique, i.e. one and only, output for every input x.
Note that in the above example if x = 2.1, then 2 is
only one integer that’s with ¾ of 2.1, i.e. y = 2. But if
a procedure produces multiple or changing number of
outputs, then we need to further process these
outputs. Think of an app on the phone that tells you all the gas stations
within 1 mile vs informing you just the closest gas station.

18. The Basic Language of Functions
b. Input a number x, the output is (are) integer(s)
within ¾ of x. Is this a function?
No, this is not a function because if x = ½, there'll be
two different outputs y = 0 or 1.
¾ ¾
1/2
–1 0 1 2
The two integers within ¾ of 1/2
For a procedure to be a function, it must produce a
unique, i.e. one and only, output for every input x.
Note that in the above example if x = 2.1, then 2 is
only one integer that’s with ¾ of 2.1, i.e. y = 2. But if
a procedure produces multiple or changing number of
outputs, then we need to further process these
outputs.

19. The Basic Language of Functions
b. Input a number x, the output is (are) integer(s)
within ¾ of x. Is this a function?
No, this is not a function because if x = ½, there'll be
two different outputs y = 0 or 1.
¾ ¾
1/2
–1 0 1 2
The two integers within ¾ of 1/2
For a procedure to be a function, it must produce a
unique, i.e. one and only, output for every input x.
Note that in the above example if x = 2.1, then 2 is
only one integer that’s with ¾ of 2.1, i.e. y = 2. But if
a procedure produces multiple or changing number of
outputs, then we need to further process these
outputs. That’s why such a procedure is not classified
as a function, which always produces single output.

20. The Basic Language of Functions
There are many ways to prescribe functions.

21. The Basic Language of Functions
There are many ways to prescribe functions.
A function may be defined by verbal descriptions such
as the [x] above.

22. The Basic Language of Functions
There are many ways to prescribe functions.
A function may be defined by verbal descriptions such
as the [x] above. A function may be given by a table
as the one shown below.
x y
–1 4
2 3
5 –3
6 4
7 2
A table–function

23. The Basic Language of Functions
There are many ways to prescribe functions.
A function may be defined by verbal descriptions such
as the [x] above. A function may be given by a table
as the one shown below. From the table we see that
for the input 2 the output is 3.
x y
–1 4
2 3
5 –3
6 4
7 2
A table–function

24. The Basic Language of Functions
There are many ways to prescribe functions.
A function may be defined by verbal descriptions such
as the [x] above. A function may be given by a table
as the one shown below. From the table we see that
for the input 2 the output is 3.
Given a function, the set D of all the valid inputs is the
domain of the function, the set R of all the outputs is
x y called the range of the function.
–1 4
2 3
5 –3
6 4
7 2
A table–function

25. The Basic Language of Functions
There are many ways to prescribe functions.
A function may be defined by verbal descriptions such
as the [x] above. A function may be given by a table
as the one shown below. From the table we see that
for the input 2 the output is 3.
Given a function, the set D of all the valid inputs is the
domain of the function, the set R of all the outputs is
x y called the range of the function.
–1 4 The domain D of the truncation function
2 3 [x] is the set of all real numbers.
5 –3
6 4
7 2
A table–function

26. The Basic Language of Functions
There are many ways to prescribe functions.
A function may be defined by verbal descriptions such
as the [x] above. A function may be given by a table
as the one shown below. From the table we see that
for the input 2 the output is 3.
Given a function, the set D of all the valid inputs is the
domain of the function, the set R of all the outputs is
x y called the range of the function.
–1 4 The domain D of the truncation function
2 3 [x] is the set of all real numbers.
5 –3 The range R of [x] is the set of all integers.
6 4
7 2
A table–function

27. The Basic Language of Functions
There are many ways to prescribe functions.
A function may be defined by verbal descriptions such
as the [x] above. A function may be given by a table
as the one shown below. From the table we see that
for the input 2 the output is 3.
Given a function, the set D of all the valid inputs is the
domain of the function, the set R of all the outputs is
x y called the range of the function.
–1 4 The domain D of the truncation function
2 3 [x] is the set of all real numbers.
5 –3 The range R of [x] is the set of all integers.
6 4 With the table–function,
the domain is D = {–1, 2, 5, 6, 7},
7 2
A table–function

28. The Basic Language of Functions
There are many ways to prescribe functions.
A function may be defined by verbal descriptions such
as the [x] above. A function may be given by a table
as the one shown below. From the table we see that
for the input 2 the output is 3.
Given a function, the set D of all the valid inputs is the
domain of the function, the set R of all the outputs is
x y called the range of the function.
–1 4 The domain D of the truncation function
2 3 [x] is the set of all real numbers.
5 –3 The range R of [x] is the set of all integers.
6 4 With the table–function,
the domain is D = {–1, 2, 5, 6, 7},
7 2
A table–function
the range is R = {4, 3, –3, 2}.

29. The Basic Language of Functions
Functions may be given graphically:

30. The Basic Language of Functions
Functions may be given graphically:
For instance, Nominal Price(1975) $0.50

31. The Basic Language of Functions
Functions may be given graphically:
For instance, Nominal Price(1975) $0.50
Domain = {year 1918  2005}

32. The Basic Language of Functions
Functions may be given graphically:
For instance, Nominal Price(1975) $1.00
Domain = {year 1918  2005}
Range (Nominal Price) = {$0.20$2.51}

33. The Basic Language of Functions
Functions may be given graphically:
Inflation Adjusted Price(1975) $1.85

34. The Basic Language of Functions
Functions may be given graphically:
Inflation Adjusted Price(1975) $1.85
Domain = {year 1918  2005}

35. The Basic Language of Functions
Functions may be given graphically:
Inflation Adjusted Price(1975) $1.85
Domain = {year 1918  2005}
Range (Inflation Adjusted Price) = {$1.25$3.50}

36. The Basic Language of Functions
Functions may be given graphically:
As the example shows here that it’s easy to
compare functions using their graphs.

37. The Basic Language of Functions
Functions may be given graphically:
As the example shows here that it’s easy to
compare functions using their graphs.
The drawback of graphs is that they are not precise.

38. The Basic Language of Functions
Functions may be given graphically:
As the example shows here that it’s easy to
compare functions using their graphs.
The drawback of graphs is that they are not precise.
To obtain accuracy we need the explicit steps for
calculating the output, i.e. mathematics formulas.

39. The Basic Language of Functions
Most functions are given by mathematics formulas.

40. The Basic Language of Functions
Most functions are given by mathematics formulas.
For example,
f(X) = X 2 – 2X + 3 = y

41. The Basic Language of Functions
Most functions are given by mathematics formulas.
For example,
f(X) = X 2 – 2X + 3 = y
name of
the function

42. The Basic Language of Functions
Most functions are given by mathematics formulas.
For example,
f(X) = X 2 – 2X + 3 = y
name of
the function We need names for
functions so we may track
multiple functions in the same
context.

43. The Basic Language of Functions
Most functions are given by mathematics formulas.
For example,
input box
f(X) = X 2 – 2X + 3 = y
name of
the function

44. The Basic Language of Functions
Most functions are given by mathematics formulas.
For example,
input box
f(X) = X 2 – 2X + 3 = y
name of
the function
The input box holds the input for the formula.

45. The Basic Language of Functions
Most functions are given by mathematics formulas.
For example,
input box
f(X) = X 2 – 2X + 3 = y
name of actual formula
the function
The input box holds the input for the formula.

46. The Basic Language of Functions
Most functions are given by mathematics formulas.
For example,
input box
f(X) = X 2 – 2X + 3 = y
name of actual formula The output
the function
The input box holds the input for the formula.

47. The Basic Language of Functions
Most functions are given by mathematics formulas.
For example,
input box
f(X) = X 2 – 2X + 3 = y
name of actual formula The output
the function
The input box holds the input for the formula.
Hence f (2) means to replace x by (2) in the formula,
so f(2) = (2)2 – 2(2) + 3

48. The Basic Language of Functions
Most functions are given by mathematics formulas.
For example,
input box
f(X) = X 2 – 2X + 3 = y
name of actual formula The output
the function
The input box holds the input for the formula.
Hence f (2) means to replace x by (2) in the formula,
so f(2) = (2)2 – 2(2) + 3 = 3 = y.

49. The Basic Language of Functions
Most functions are given by mathematics formulas.
For example,
input box
f(X) = X 2 – 2X + 3 = y
name of actual formula The output
the function
The input box holds the input for the formula.
Hence f (2) means to replace x by (2) in the formula,
so f(2) = (2)2 – 2(2) + 3 = 3 = y.
Let’s take a closer look at some more examples

50. The Basic Language of Functions
Example B. Let f(x) = –3x + 2, g(x) = –2x2 – 3.
a. Evaluate f(–2)

51. The Basic Language of Functions
Example B. Let f(x) = –3x + 2, g(x) = –2x2 – 3.
a. Evaluate f(–2)
f(x) = –3x + 2

52. The Basic Language of Functions
Example B. Let f(x) = –3x + 2, g(x) = –2x2 – 3.
a. Evaluate f(–2)
f(x) = –3x + 2
f(–2)

53. The Basic Language of Functions
Example B. Let f(x) = –3x + 2, g(x) = –2x2 – 3.
a. Evaluate f(–2)
f(x) = –3x + 2
f(–2)
copy the input

54. The Basic Language of Functions
Example B. Let f(x) = –3x + 2, g(x) = –2x2 – 3.
a. Evaluate f(–2)
f(x) = –3x + 2
f(–2) = –3(–2) + 2
then paste the input
copy the input at where the x’s are

55. The Basic Language of Functions
Example B. Let f(x) = –3x + 2, g(x) = –2x2 – 3.
a. Evaluate f(–2)
f(x) = –3x + 2
f(–2) = –3(–2) + 2 = 8
then paste the input
copy the input at where the x’s are

56. The Basic Language of Functions
Example B. Let f(x) = –3x + 2, g(x) = –2x2 – 3x + 1.
a. Evaluate f(–2).
f(–2) = –3(–2) + 2 = 8

57. The Basic Language of Functions
Example B. Let f(x) = –3x + 2, g(x) = –2x2 – 3x + 1.
a. Evaluate f(–2).
f(–2) = –3(–2) + 2 = 8
b. Evaluate g(–2).

58. The Basic Language of Functions
Example B. Let f(x) = –3x + 2, g(x) = –2x2 – 3x + 1.
a. Evaluate f(–2).
f(–2) = –3(–2) + 2 = 8
b. Evaluate g(–2).
g(x) = –2x 2 – 3x + 1

59. The Basic Language of Functions
Example B. Let f(x) = –3x + 2, g(x) = –2x2 – 3x + 1.
a. Evaluate f(–2).
f(–2) = –3(–2) + 2 = 8
b. Evaluate g(–2).
g(x) = –2x 2 – 3x + 1
g(–2)

60. The Basic Language of Functions
Example B. Let f(x) = –3x + 2, g(x) = –2x2 – 3x + 1.
a. Evaluate f(–2).
f(–2) = –3(–2) + 2 = 8
b. Evaluate g(–2).
g(x) = –2x 2 – 3x + 1
g(–2)
copy the input

61. The Basic Language of Functions
Example B. Let f(x) = –3x + 2, g(x) = –2x2 – 3x + 1.
a. Evaluate f(–2).
f(–2) = –3(–2) + 2 = 8
b. Evaluate g(–2).
g(x) = –2x 2 – 3x + 1
g(–2) = –2(–2)2 – 3(–2) + 1
then paste the input at where
copy the input
the x’s are in the formula.

62. The Basic Language of Functions
Example B. Let f(x) = –3x + 2, g(x) = –2x2 – 3x + 1.
a. Evaluate f(–2).
f(–2) = –3(–2) + 2 = 8
b. Evaluate g(–2).
g(–2) = –2(–2)2 – 3(–2) + 1

63. The Basic Language of Functions
Example B. Let f(x) = –3x + 2, g(x) = –2x2 – 3x + 1.
a. Evaluate f(–2).
f(–2) = –3(–2) + 2 = 8
b. Evaluate g(–2).
g(–2) = –2(–2)2 – 3(–2) + 1
= –8 + 6 + 1 = –1

64. The Basic Language of Functions
Example B. Let f(x) = –3x + 2, g(x) = –2x2 – 3x + 1.
a. Evaluate f(–2).
f(–2) = –3(–2) + 2 = 8
b. Evaluate g(–2).
g(–2) = –2(–2)2 – 3(–2) + 1
= –8 + 6 + 1 = –1
We may +, –, *, / functions.

65. The Basic Language of Functions
Example B. Let f(x) = –3x + 2, g(x) = –2x2 – 3x + 1.
a. Evaluate f(–2).
f(–2) = –3(–2) + 2 = 8
b. Evaluate g(–2).
g(–2) = –2(–2)2 – 3(–2) + 1
= –8 + 6 + 1 = –1
We may +, –, *, / functions.
c. Evaluate f(–2) – g(–2).

66. The Basic Language of Functions
Example B. Let f(x) = –3x + 2, g(x) = –2x2 – 3x + 1.
a. Evaluate f(–2).
f(–2) = –3(–2) + 2 = 8
b. Evaluate g(–2).
g(–2) = –2(–2)2 – 3(–2) + 1
= –8 + 6 + 1 = –1
We may +, –, *, / functions.
c. Evaluate f(–2) – g(–2).
From parts a and b we’ve
f(–2) – g(–2) = 8 – (–1)

67. The Basic Language of Functions
Example B. Let f(x) = –3x + 2, g(x) = –2x2 – 3x + 1.
a. Evaluate f(–2).
f(–2) = –3(–2) + 2 = 8
b. Evaluate g(–2).
g(–2) = –2(–2)2 – 3(–2) + 1
= –8 + 6 + 1 = –1
We may +, –, *, / functions.
c. Evaluate f(–2) – g(–2).
From parts a and b we’ve
f(–2) – g(–2) = 8 – (–1) = 9

68. The Basic Language of Functions
Example B. Let f(x) = –3x + 2, g(x) = –2x2 – 3x + 1.
a. Evaluate f(–2).
f(–2) = –3(–2) + 2 = 8
b. Evaluate g(–2).
g(–2) = –2(–2)2 – 3(–2) + 1
= –8 + 6 + 1 = –1
We may +, –, *, / functions.
c. Evaluate f(–2) – g(–2).
From parts a and b we’ve
f(–2) – g(–2) = 8 – (–1) = 9
The function f(x) = c where c is a number is a constant
function.

69. The Basic Language of Functions
Example B. Let f(x) = –3x + 2, g(x) = –2x2 – 3x + 1.
a. Evaluate f(–2).
f(–2) = –3(–2) + 2 = 8
b. Evaluate g(–2).
g(–2) = –2(–2)2 – 3(–2) + 1
= –8 + 6 + 1 = –1
We may +, –, *, / functions.
c. Evaluate f(–2) – g(–2).
From parts a and b we’ve
f(–2) – g(–2) = 8 – (–1) = 9
The function f(x) = c where c is a number is a constant
function. The output of a constant function does not
change. So if g(x) = 3 then g(0) = g(5) = 3.

70. The Basic Language of Functions
When a function is given by a math–formula f, its
domain, i.e. the legitimate inputs for f, needs to be
determined first.

71. The Basic Language of Functions
When a function is given by a math–formula f, its
domain, i.e. the legitimate inputs for f, needs to be
determined first. There are two main considerations
when determining the domain of a given formula f.

72. The Basic Language of Functions
When a function is given by a math–formula f, its
domain, i.e. the legitimate inputs for f, needs to be
determined first. There are two main considerations
when determining the domain of a given formula f.
1. The denominators in f can't be 0.

73. The Basic Language of Functions
When a function is given by a math–formula f, its
domain, i.e. the legitimate inputs for f, needs to be
determined first. There are two main considerations
when determining the domain of a given formula f.
1. The denominators in f can't be 0.
2. The radicands of the square roots in f (or any even
root) must be non–negative.

74. The Basic Language of Functions
When a function is given by a math–formula f, its
domain, i.e. the legitimate inputs for f, needs to be
determined first. There are two main considerations
when determining the domain of a given formula f.
1. The denominators in f can't be 0.
2. The radicands of the square roots in f (or any even
root) must be non–negative.
Example C. Find the domain of the following functions.
a. f(x) = 1/(2x + 6)
b. f (X) = 2x + 6

75. The Basic Language of Functions
When a function is given by a math–formula f, its
domain, i.e. the legitimate inputs for f, needs to be
determined first. There are two main considerations
when determining the domain of a given formula f.
1. The denominators in f can't be 0.
2. The radicands of the square roots in f (or any even
root) must be non–negative.
Example C. Find the domain of the following functions.
a. f(x) = 1/(2x + 6)
The denominator can’t be 0, i.e. 2x + 6 ≠ 0
b. f (X) = 2x + 6

76. The Basic Language of Functions
When a function is given by a math–formula f, its
domain, i.e. the legitimate inputs for f, needs to be
determined first. There are two main considerations
when determining the domain of a given formula f.
1. The denominators in f can't be 0.
2. The radicands of the square roots in f (or any even
root) must be non–negative.
Example C. Find the domain of the following functions.
a. f(x) = 1/(2x + 6)
The denominator can’t be 0, i.e. 2x + 6 ≠ 0  x ≠ –3
b. f (X) = 2x + 6

77. The Basic Language of Functions
When a function is given by a math–formula f, its
domain, i.e. the legitimate inputs for f, needs to be
determined first. There are two main considerations
when determining the domain of a given formula f.
1. The denominators in f can't be 0.
2. The radicands of the square roots in f (or any even
root) must be non–negative.
Example C. Find the domain of the following functions.
a. f(x) = 1/(2x + 6)
The denominator can’t be 0, i.e. 2x + 6 ≠ 0  x ≠ –3
So the domain = {all numbers except x = –3}.
b. f (X) = 2x + 6

78. The Basic Language of Functions
When a function is given by a math–formula f, its
domain, i.e. the legitimate inputs for f, needs to be
determined first. There are two main considerations
when determining the domain of a given formula f.
1. The denominators in f can't be 0.
2. The radicands of the square roots in f (or any even
root) must be non–negative.
Example C. Find the domain of the following functions.
a. f(x) = 1/(2x + 6)
The denominator can’t be 0, i.e. 2x + 6 ≠ 0  x ≠ –3
So the domain = {all numbers except x = –3}.
b. f (X) = 2x + 6
We must have 2x + 6 > 0 to extract its square root,

79. The Basic Language of Functions
When a function is given by a math–formula f, its
domain, i.e. the legitimate inputs for f, needs to be
determined first. There are two main considerations
when determining the domain of a given formula f.
1. The denominators in f can't be 0.
2. The radicands of the square roots in f (or any even
root) must be non–negative.
Example C. Find the domain of the following functions.
a. f(x) = 1/(2x + 6)
The denominator can’t be 0, i.e. 2x + 6 ≠ 0  x ≠ –3
So the domain = {all numbers except x = –3}.
b. f (X) = 2x + 6
We must have 2x + 6 > 0 to extract its square root,
or that x > –3.

80. The Basic Language of Functions
When a function is given by a math–formula f, its
domain, i.e. the legitimate inputs for f, needs to be
determined first. There are two main considerations
when determining the domain of a given formula f.
1. The denominators in f can't be 0.
2. The radicands of the square roots in f (or any even
root) must be non–negative.
Example C. Find the domain of the following functions.
a. f(x) = 1/(2x + 6)
The denominator can’t be 0, i.e. 2x + 6 ≠ 0  x ≠ –3
So the domain = {all numbers except x = –3}.
b. f (X) = 2x + 6
We must have 2x + 6 > 0 to extract its square root,
or that x > –3. So the domain = [–3, ∞) .

81. The Basic Language of Functions
Graphs of Functions

82. The Basic Language of Functions
Graphs of Functions
The plot of all points (x, y)’s
that satisfy a given function
y = f(x) is the graph of the
function y = f(x).

83. The Basic Language of Functions
Graphs of Functions
The plot of all points (x, y)’s
that satisfy a given function
y = f(x) is the graph of the
function y = f(x).
For example, let the function
f(x) = x + 1,

84. The Basic Language of Functions
Graphs of Functions
The plot of all points (x, y)’s
that satisfy a given function
y = f(x) is the graph of the
function y = f(x).
For example, let the function
f(x) = x + 1, set y = f(x) and
make a table of few of the
ordered pairs (x, y)’s that
satisfy the equation y = f(x).

85. The Basic Language of Functions
Graphs of Functions
The plot of all points (x, y)’s x 0 1 2 3
that satisfy a given function
y = f(x) 1 2 3 4
y = f(x) is the graph of the
function y = f(x).
For example, let the function
f(x) = x + 1, set y = f(x) and
make a table of few of the
ordered pairs (x, y)’s that
satisfy the equation y = f(x).

86. The Basic Language of Functions
Graphs of Functions
The plot of all points (x, y)’s x 0 1 2 3
that satisfy a given function
y = f(x) 1 2 3 4
y = f(x) is the graph of the
function y = f(x).
For example, let the function
f(x) = x + 1, set y = f(x) and
make a table of few of the
ordered pairs (x, y)’s that
satisfy the equation y = f(x).
Plot the (x, y)’s and we have the graph of f(x) = x + 1,
a line.

87. The Basic Language of Functions
Graphs of Functions
The plot of all points (x, y)’s x 0 1 2 3
that satisfy a given function
y = f(x) 1 2 3 4
y = f(x) is the graph of the
function y = f(x). y=x+1
For example, let the function
f(x) = x + 1, set y = f(x) and
make a table of few of the
ordered pairs (x, y)’s that
satisfy the equation y = f(x).
Plot the (x, y)’s and we have the graph of f(x) = x + 1,
a line.

88. The Basic Language of Functions
Graphs of Functions
The plot of all points (x, y)’s x 0 1 2 3
that satisfy a given function
y = f(x) 1 2 3 4
y = f(x) is the graph of the
function y = f(x). y=x+1
For example, let the function
f(x) = x + 1, set y = f(x) and
make a table of few of the
ordered pairs (x, y)’s that
satisfy the equation y = f(x).
Plot the (x, y)’s and we have the graph of f(x) = x + 1,
a line. Note that the graph of a function may cross any
vertical line at most at one point because for each x
there is only one corresponding output y.

89. The Basic Language of Functions
Graphs of Functions
The plot of all points (x, y)’s x 0 1 2 3
that satisfy a given function
y = f(x) 1 2 3 4
y = f(x) is the graph of the
function y = f(x). y=x+1
For example, let the function
f(x) = x + 1, set y = f(x) and
make a table of few of the
ordered pairs (x, y)’s that
satisfy the equation y = f(x).
Plot the (x, y)’s and we have the graph of f(x) = x + 1,
a line. Note that the graph of a function may cross any
vertical line at most at one point because for each x
there is only one corresponding output y.

90. The Basic Language of Functions
The equation x = y2, treating x
as the input, is not a function.

91. The Basic Language of Functions
The equation x = y2, treating x
as the input, is not a function.
For instance, for the input
x = 4, there are two output
y’s that satisfy 4 = y2,

92. The Basic Language of Functions
The equation x = y2, treating x
as the input, is not a function.
For instance, for the input
x = 4, there are two output
y’s that satisfy 4 = y2, namely
y = 2 and y = –2.

93. The Basic Language of Functions
The equation x = y2, treating x
as the input, is not a function.
For instance, for the input
x = 4, there are two output
y’s that satisfy 4 = y2, namely
y = 2 and y = –2. So x = y2 is
not a function.

94. The Basic Language of Functions
The equation x = y2, treating x x 0 1 1 4 4
as the input, is not a function. y 0 1 -1 2 -2
For instance, for the input
x = 4, there are two output
y’s that satisfy 4 = y2, namely
y = 2 and y = –2. So x = y2 is
not a function.
Plot the graph by the table
shown.

95. The Basic Language of Functions
The equation x = y2, treating x x 0 1 1 4 4
as the input, is not a function. y 0 1 -1 2 -2
For instance, for the input
x = 4, there are two output
y’s that satisfy 4 = y2, namely x = y2
y = 2 and y = –2. So x = y2 is
not a function. x 0 1 1 4 4
Plot the graph by the table y 0 1 -1 2 -2
shown.

96. The Basic Language of Functions
The equation x = y2, treating x x 0 1 1 4 4
as the input, is not a function. y 0 1 -1 2 -2
For instance, for the input
x = 4, there are two output
y’s that satisfy 4 = y2, namely x = y2
y = 2 and y = –2. So x = y2 is
not a function. x 0 1 1 4 4
Plot the graph by the table y 0 1 -1 2 -2
shown. In particular that if we
draw the vertical line x = 4,
it intersects the graph at two points (4, 2) and (4, –2).

97. The Basic Language of Functions
The equation x = y2, treating x x 0 1 1 4 4
as the input, is not a function. y 0 1 -1 2 -2
For instance, for the input
x = 4, there are two output
y’s that satisfy 4 = y2, namely x = y2
y = 2 and y = –2. So x = y2 is
not a function. x 0 1 1 4 4
Plot the graph by the table y 0 1 -1 2 -2
shown. In particular that if we
draw the vertical line x = 4,
it intersects the graph at two points (4, 2) and (4, –2).
In general if any vertical line crosses a graph at two
or more points then the graph does not represent a
function.

98. The Basic Language of Functions
Since for functions each
input x has exactly one
output, therefore each
vertical line can only
intersect it’s graph at exactly
one location (e.g. y = x + 1).

99. The Basic Language of Functions
Since for functions each
input x has exactly one y=x+1
output, therefore each
vertical line can only
intersect it’s graph at exactly
one location (e.g. y = x + 1).

100. The Basic Language of Functions
Since for functions each
input x has exactly one y=x+1
output, therefore each
vertical line can only
intersect it’s graph at exactly
one location (e.g. y = x + 1).
However, if any vertical line
intersects a graph at two or
more points, i.e. there are two
or more outputs y associated
to one input x (eg. x = y2),

101. The Basic Language of Functions
Since for functions each
input x has exactly one y=x+1
output, therefore each
vertical line can only
intersect it’s graph at exactly
one location (e.g. y = x + 1).
However, if any vertical line
intersects a graph at two or
more points, i.e. there are two
or more outputs y associated
to one input x (eg. x = y2),
then the graph must not be
the graph of a function.

102. The Basic Language of Functions
Since for functions each
input x has exactly one y=x+1
output, therefore each
vertical line can only
intersect it’s graph at exactly
one location (e.g. y = x + 1).
However, if any vertical line
intersects a graph at two or
more points, i.e. there are two
or more outputs y associated
to one input x (eg. x = y2), x 0 1 1 4 4
then the graph must not be y 0 1 -1 2 -2
the graph of a function.