Functions and their Graphs

Contributed by:

Sharp Tutor Fri, Jan 14, 2022 10:24 PM UTC

Objective To be able to identify a function and to be able to graphically represent functions. Purpose To help describe input-output relations in real-world applications and to use functions to model and solve real-life problems.

1. Functions and their Graphs
Lesson 2
Sections 1.2 and 1.3

2. Directions for Open–Minded
Questions – Warm Up
 Everyone silently read through both problems.
 Write down what you know.
 Read the two questions again.
 Group Discuss
 You have 5 minutes to read and discuss and 10 minutes or less to show solutions.

3. Solve the following. Try to find
multiple solution paths!
 In an all-adult apartment building, 2/3 of the men are
married to 3/5 of the women. What fraction of the
residents is married?
 A farmer had hens and rabbits. These animals have
50 heads and 140 feet. How many hens and rabbits
does the farmer have?

4. In an all-adult apartment building, 2/3 of the men are married
to 3/5 of the women. What fraction of the residents is married?

5. A farmer had hens and rabbits. These animals have 50 heads
and 140 feet. How many hens and rabbits does the farmer have?

6.  Objective
To be able to identify a function and to be
able to graphically represent functions.
 Purpose
To help describe input-output relations in
real-world applications and to use functions
to model and solve real-life problems.

7.  Relation – pairs of quantities that are related
to each other
 Example: The area A of a circle is related to
its radius r by the formula
2
A r .

8.  There are different kinds of relations.
 When a relation matches each item from one
set with exactly one item from a different set
the relation is called a function.

9. Definition of a Function
 A function is a relationship between two
variables such that each value of the first
variable is paired with exactly one value of
the second variable.
 The domain is the set of permitted x values.
 The range is the set of found values of y.
These can be called images.

10. Is it a Function?
 For each x, there is
only one value of y. Domain, x Range, y
1 -3.6
 Therefore, it IS a 2 -3.6
function. 3 4.2
4 4.2
5 10.7
6 12.1
52 52

11. Is it a function?
 Three different y-
values (7, 8, and 10) Domain, x Range, y
are paired with one x- 3 7
value. 3 8
3 10
 Therefore, it is NOT a
4 42
function
10 34
11 18
52 52

12.  Is it a function? State the domain and range.
{(5, 8), (6, 7), (3, -1), (4, 2), (5, 9), (12, -2)
 No. The x-value of 5 is paired with two
different y-values.
 Domain: (5, 6, 3, 4, 12)
 Range: (8, 7, -1, 2, 9, -2)

13. Vertical Line Test
 Used to determine if a graph is a function.
 If a vertical line intersects the graph at more than
one point, then the graph is NOT a function.
NOT a Function

14. Is it a function? Give the domain
and range.
FUNCTION
Domain :   4,2
Range :   4,4

15. Give the Domain and Range.
Domain : x 1 Domain :  2  x 2
Range : y 2 Range : 0  y 3

16. Functional Notation
 We have seen an equation written in the form
y = some expression in x.
 Another way of writing this is to use
functional notation.
 For Example, you could write y = x²
as f(x) = x².

17. Functional Notation: Find the
f ( x ) 3 x  x  2
2
f ( x) x  x  22
f ( 3) f ( m  3)
2
2
3  3    3  2  m  3   m  3  2
27  3  2  m  3 m  3  m  3  2
30  2
m 2  3m  3m  9  m  3  2
m 2  5m  8

18. Let’s look at Functions

19. Find: f ( 2)  g ( 4)
f ( x) g ( x)

20. Find: f ( 5)  g ( 0 )
f ( x) g ( x)

21. Find: f ( 4) g (  1)
f ( x) g ( x)

22. Find: f (  2)  g ( 0)
f ( x) g ( x)

23.

24.  A piecewise-defined function is a function that is
defined by two or more equations over a specified
domain.
 The absolute value function f  x   x
can be written as a piecewise-defined function.
 The basic characteristics of the absolute value
function are summarized on the next page.

25. Absolute Value Function is a
Piecewise Function

26.  Evaluate the function when x = -1 and 0.

27. Domain of a Function

28.  The domain of a function can be implied by
the expression used to define the function
 The implied domain is the set of all real
numbers for which the expression is defined.
 For example,

29.  The function has an implied
domain that consists of all real x other than
x = ±2
 The domain excludes x-values that result
in division by zero.

30.  Another common type of implied domain is
that used to avoid even roots of negative
numbers.
 EX:
is defined only for x 0.
The domain excludes x-values that result
in even roots of negative numbers.

31. Graphs of Functions
Lesson 3

32.  Objective:
To graph a function using domain and
range, even or odd, relative min/max.
 Purpose:
To introduce methods to help graph a
function.

33. Domain & Range of a Function
What is the
domain of
the graph of
the function
f?
A :   1,4 

34. Domain & Range of a Function
What is the
range of
the graph of
the function
f?
  5,4

35. Domain & Range of a Function
Find f   1 and f  2 .
f   1   5
f  2  4

36.  Let’s look at domain and range of a
function using an algebraic approach.
 Then, let’s check it with a graphical
approach.

37. Find the domain and range of
f  x   x  4.
 Algebraic Approach
The expression under the radical can not be negative.
Therefore, x  4 0. Domain
A : x 4 Since the domain is never negative the
or range is the set of all nonnegative real
 4,  numbers.
A : y 0
or Range
 0, 

38. Find the domain and range of
f  x   x  4.
 Graphical Approach

39. Increasing and Decreasing

40.  The more you know about the graph of
a function, the more you know about
the function itself.
 Consider the graph on the next slide.

41. Falls from x = -2 to x = 0.
Is constant from x = 0 to
x = 2.
Rises from x = 2 to x = 4.

42. Ex: Find the open intervals on which the
function is increasing, decreasing, or constant.
Increases over
the entire real
line.

43. Ex: Find the open intervals on which the
function is increasing, decreasing, or constant.
INCREASING :
  , 1 and 1, 
DECREASING :
  1,1

44. Ex: Find the open intervals on which the
function is increasing, decreasing, or constant.
INCREASING :
  ,0
CONSTANT :
 0,2
DECREASING :
 2, 

45. Relative Minimum and
Maximum Values

46. Relative Min/Max
 The point at which a function changes
its increasing, decreasing, or constant
behavior are helpful in determining the
relative maximum or relative
minimum values of a function.

47. General Points – We’ll find
EXACT points later……

48. Approximating a Relative
 Example: Use a GDC to approximate
the relative minimum of the function
given by
2
f  x  3 x  4 x  2.

49. 2
f  x  3 x  4 x  2 .
 Put the function into the “y = “ the
press zoom 6 to look at the graph.
 Press 2nd Calc, 3:minimum, left bound,
right bound, enter at the lowest point.

50.  Use a GDC to approximate the relative
minimum and relative maximum of the
function given by
3
f  x   x  x.

51. Relative Minimum
(-0.58, -0.38)

52. Relative Maximum
(0.58, 0.38)

53. Step Functions and
Piecewise-Defined Functions

54. Because of the vertical jumps, the greatest integer function is an example
of a step function.

55. Let’s graph a Piecewise-
Defined Function
 Sketch the graph of
 2 x  3, x 1
f  x  
 x  4, x  1
Notice when open
dots and closed
dots are used. Why?

56. Even and Odd Functions

57.

58. Algebraically
Let’s look at the graphs again and see if this applies.

59. ☺ ☺

60.  Determine whether each function is
even, odd, or neither.

61. Graphical –
Algebraic Symmetric to
Origin

62. Graphical –
Symmetric to y-
axis

63. Graphical – NOT
Algebraic Symmetric to
origin OR y-axis.

64. You Try
 Is the function
f  x  x
 Even, Odd, of Neither?

65. f  x  x
Symmetric about the y-axis.

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