This document defines and provides examples of various types of functions including: constant functions, identity functions, polynomial functions, absolute value functions, square root functions, piecewise functions, greatest integer functions, least integer functions, increasing and decreasing functions, and shifting graphs of functions. It also discusses domain and range, composite functions, trigonometric functions, exponential functions, and logarithmic functions.

1. • Be able to understand:
• What is a constant function?
• What is an Identity function?
• What is polynomial function?
• What is an absolute function?
• What is square root function?
• What is piecewise function?
• What is greatest integer function?
• What is least integer function?
• What is increasing and decreasing
function?
Dr Sumaira Rehman

10. Draw the graph of the function
y=𝟏 + 𝒙𝟐
And write the domain and range ?
Dr Sumaira Rehman

13. Small Exercise about shifting the absolute function

15. Write the domain and range of the function
y= 𝑥 − 3
Write the domain and range of the function
y=1- 𝒙

17. A piecewise function is a function where more than one formula is used
to define the output. Each formula has its own domain, and the domain of
the function is the union of all of these smaller domains.
Piecewise function:

18. Example 1:

20. Example 2:

26. Least integer function (Ceiling function)
Also known as the ceiling function, LIF(x) is the least
integer function, which returns the value of the least
integer more than or equal to x. For example, LIF(3.55)
will return a value 4.
For example
⌈3.578⌉ = 4
⌈0.78⌉ = 1
⌈-4.64⌉ = - 4

31. Dr
Sumaira
Rehman

40. 3. Composite function :
When the output from one function is used as the input to another function we
form what is known as a composite function.

41. FIGURE 3.1 : Arrow diagram for ƒ o g.

42. Example 3.1

44. Here the function rule is 'multiply the input by 3 and then add
2'. The composite
function f (f (x)) is illustrated in Figure 3.2.
Example 3.2
Solution
Figure 3.2 The composition of f (x) with itself.

47. Shifting the graph of the function:

51. To shift the graph ƒ(x)=𝒙𝟐 of up (or down), we add
positive (or negative) constants to the formula for ƒ

52. To shift the graph of to the y=𝒙𝟐 left, we add a positive constant to x.
To shift the graph to the right, we add a negative constant to x

53. Adding -2 to x in y= 𝐱 and then -1 adding to the result, gives y= 𝐱 − 𝟐 − 𝟏
and shifts the graph 2 units to the right and 1 unit down
Shifting the graph of the function y= 𝐱

64. The trigonometric functions are important because they are periodic, or
repeating, and therefore model many naturally occurring periodic processes.

79. Exponential Functions

82. Exponential functions have the form f(x) = bx, where b > 0
and b ≠ 1. Just as in any exponential expression, b is called
the base and x is called the exponent.

83. An example of an exponential function is the growth of bacteria.
Some bacteria double every hour. If you start with 1 bacterium
and it doubles every hour, you will have 2x bacteria
after x hours. This can be written as f(x) = 2x.
Example
With the definition f(x) = bx and the restrictions that b > 0 and that b ≠ 1, the domain of
an exponential function is the set of all real numbers. The range is the set of all positive
real numbers. The following graph shows f(x) = 2x.

86. Logarithmic Functions
These are the functions 𝐟 𝐱 = 𝐥𝐨𝐠𝐚 𝐱 where the base 𝐚 ≠ 𝟏
is a positive constant. They are the inverse functions of the
exponential functions.