Unit 1 Day 4
Algebra 2
Name:______________
Inverses, End Behaviors and
Restricted Domains
Today:
 You will analyze and find the inverse of a function. You will also determine the domain of the
function and its inverse. You will also evaluate the end behaviors of functions.
 You will review functions and explore domain and range with asymptotes and holes.
 At the end of class, you will be able to find an inverse and determine whether the inverse is a
function. You will also be able to discuss end behaviors and domain and range of continuous and
discontinuous graphs.
Warm-Up: State the domain and range for each of the following relations and state whether or not the
relation is a function.
1) 3,2, 5,2,  6,2
2)
3)
4
12
5
15
8
10
y
4) Using the graph to the right, where is the graph increasing?
x
Where is the graph decreasing?
5) Identify the local maximum and minimum of the graph below:
6) Name the property: 4 + (-4) = 0
Review: For each of the following state the interval(s) for the domain and range, the interval(s) in which
the function is increasing, decreasing, and constant.
Sketch a graph of a function with the given domain and range. Graphs will vary.
b. Domain : (-, -2]  [1, )
Range: [3, )
a. Domain: [-3, 2]
Range: [-6, 3]
y
y
y
x
c. Domain: (-, )
Range: (-, )
x
x
Inverse Functions: To find the inverse of a function, you will reflect the function over the y = x line.
The easiest way to do this is to just switch the ordered pairs.
Example: Find the inverse of the following and then determine if the inverse is a function:
1. {(5, -2), (6, 3), (9, -4)}
2. {(2, 2), (3, -2), (-2, 3), (-4, 2), (0, 4)}
Graphs of Inverses__________________________________________
1. Graph f ( x)  x  2 and its inverse on the same coordinate plane.
x
y
Horizontal Line Test:
2. Graph the inverse of the following graph using the original graph:
Original:
x
Inverse:
y
x
y
End Behaviors: Graphs are pictures that show you how one thing changes in relation to another. Learning
to read graphs properly is a matter of interpreting which pieces of information go together. End Behavior
of graphs describes the ____________ of both sides of the function. You will always have ___________
answers when describing the end behavior.
Graphs are either "up" on both ends, "down" on both ends, or functions can also have ends that head off in
opposite directions. Listed below are several examples.
When describing the end behavior, you will describe each side in terms of: as x
-∞ , where does f(x)
head? You are describing the _________ side of the graph. Then you will describe as x
∞ , where
does f(x) head? You are describing the _________ side of the graph.
Practice:
as x   , f ( x)  ____
as x   , f ( x)  ____
as x   , f ( x)  ____
as x  ,
as x  ,
as x  ,
f ( x)  ____
***sometimes written f(x)
Video:
∞, as x
∞
f ( x)  ____
f ( x)  ____
Continuous Functions
Definition:
Example:
How can you determine if function is continuous?
Define a discontinuous function:
Example:
Restricted Domain and Range: If a function is not everywhere continuous, it could have a restricted
domain or range. Domain restrictions come in two basic types: vertical asymptotes and holes. Range
restrictions will come from holes and horizontal asymptotes.
Holes: Sometimes functions have a discontinuity that shows itself with a hole in the graph. The domains
for these functions are represented the same way as the domains of functions with vertical asymptotes. The
picture might look different, but the fact that there is a value or values of x that will make the function
undefined remains the same.
Ex. 1: Determine the domain and range of the functions represented by the graphs below.
a)
b)
Domain:________________________
Domain:________________________
Range:_________________________
Range:_________________________
_
_
Vertical Asymptotes: A vertical asymptote occurs whenever there is a value of x in the denominator of a
rational function that makes the denominator zero. Remember, you are never allowed to divide by zero!
Ex 2: f ( x) 
1
x2
The graph of this rational function looks like this:
The domain of this function is all real numbers EXCEPT 2 ,
which is where the function is undefined. This is the
______________________ asymptote.
There are several ways to write this domain:
: x  2
f ( x) : x  2
 ,2  2, 
x  , 2 ,  2,  
What is the Range of the function?_____________________________________________________
Ex. 3: Use the graph of the function f ( x) 
3
to write the domain and range of the function.
x3
Domain:_____________________
Range:______________________
Ex.4: Use the graph of the function or the function rule to determine its domain. Then, state the range.
g ( x) 
1
x  2x  8
2
Domain:______________________
Range:_______________________
In your own words below, describe a continuous function and a discontinuous function. Sketch one of
each.
RESTRICTED DOMAINS PRACTICE
Look for a variable in the denominator or a variable inside a square root symbol.
• The values that make the denominator zero must be excluded from the domain.
Think: Denominator ≠ 0
• The expression inside the square root symbol must be greater than or equal to zero.
Think: Radicand > 0
If there are no restrictions, then the domain of the functions (linear, quadratic, cubic, etc.) is
all real numbers, or  ,  
State the domain of each function.
1. f(x) = 5x + 17
4. v(x) = -2|x| – 8
2. j ( x) 
1
x7
3. m( x) 
5. p( x) 
3
(4 x  1)( x  9)
6. g ( x)  x  2
2
( x)( x  6)
Inverse Practice!
Graph the inverse of each of the following relations. Then, state if the inverse is a function or not.
1)
2)