1. Functions and Their Properties II
Lesson 6.2
Functions and Their Properties II
(y – day)

2. Lesson Objectives At the end of the lesson, students can:
Define and determine the range of a function or relation.
Define and determine the boundedness of a function.
Define and determine the extrema of a function.
Define and determine the limits of a function.
Define and determine the end behavior of a function.

3. Range
Find the range of the functions by looking for all y-coordinates that correspond to points on the graph:
𝑓 𝑥 = 𝑥 R:
𝑔 𝑥 =− 𝑥 −4 R:
ℎ 𝑠 = 𝑠 3 R:
ℎ 𝜃 = sin 𝜃 R:
[3,∞)
(−∞, 4]
(−∞,∞)
[−1, 1]

4. Bounded Bounded Below: Bounded Above: Bounded:
A function is bounded below if there exists some number, b, less than or equal to every number in the range of f. b is the lower bound.
A function is bounded above if there exists some number, b, greater than or equal to every number in the range of f. b is the upper bound.
A function is bounded if it is bounded both above and below.

5. Bounded Determine if the function is bounded: 𝑓 𝑥 = 𝑥 2 +3 R:
ℎ 𝜃 = sin 𝜃 R:
Bounded Below
Bounded Above
Not Bounded
Bounded

6. Extrema
Relative Maximum: Relative Minimum: Given 𝑓 𝑥 =2 𝑥 4 −5 𝑥 2 +2𝑥, find all extrema.
A relative (local) maximum of f is a value f(c) that is greater than or equal to all range values of f on some open interval containing c. If f(c) is greater than or equal to all range values, then f(c) is the max.
A relative (local) maximum of f is a value f(c) that is less than or equal to all range values of f on some open interval containing c. If f(c) is less than or equal to all range values, then f(c) is the min.

7. Extrema
Given 𝑓 𝑥 = 𝑥 4 −7 𝑥 2 +6𝑥, find all extrema. Local Max: Local Min:

8. Limits 𝑓 𝑥 = 𝑥 2 x[-5, 5] y[-1, 9] As x increases, what happens to y?
lim 𝑥→+∞ 𝑓(𝑥) = or lim 𝑥→+∞ 𝑥 2 =
As x decreases, what happens to y?
lim 𝑥→−∞ 𝑓 𝑥 =
As x approaches 0, what happens to y?
lim 𝑥→0 𝑓 𝑥 =

9. Limits 𝑓 𝑥 = 𝑥 3 x[-5, 5] y[-5, 5] As x increases, what happens to y?
lim 𝑥→+∞ 𝑓(𝑥) = or lim 𝑥→+∞ 𝑥 3 =
As x decreases, what happens to y?
lim 𝑥→−∞ 𝑓 𝑥 =
As x approaches 0, what happens to y?
lim 𝑥→0 𝑓 𝑥 =

10. End Behavior 𝑓 𝑥 = 2 𝑥 2 −1 𝑥 2 +3 x[-5, 5] y[-5, 5] lim 𝑥→+∞ 𝑓(𝑥) =

11. Homework
Read pp
Do p 98: 17, 19, 25, 27, 37 – 43 ODD

12. Lesson Objectives At the end of the lesson, students can:
Define and determine the range of a function or relation.
Define and determine the boundedness of a function.
Define and determine the extrema of a function.
Define and determine the limits of a function.
Define and determine the end behavior of a function.