Session 11
Agenda:
• Questions from 5.4-5.6?
• 6.1 – Piecewise-Defined Functions
• 6.2 – Operations on Functions
• 6.3 – One-to-One Functions and Inverses
• Things to do before our next meeting.
Questions?
6.1 – Piecewise-Defined Functions
• A piecewise-defined function is one in which the function
is defined separately on separate parts of its domain. For
example:
2
• Evaluate:
f (  5)
f (  1)
f (0 )
f (2 )
f (4 )
1  x

f (x)   x  4

 x
if
x  1
if  1  x  2
if
x2
• In the previous function, the domain is all real numbers.
Why?
• First, note that the three inequalities, x<-1, -1≤x<2, and
x≥2 make up all possible values of x.
• Secondly, notice that each function piece is defined for all
the x-values in that piece’s domain.
• Although x is not defined for negative values of x,
notice that we only use this piece of the function when
x≥2.
2
1  x

f (x)   x  4

 x
if
x  1
if  1  x  2
if
x2
Determine the domain of the following piecewise function.
 1
if
x  2
x4

 x  1 if  2  x  3
f ( x)  
2
4 x5
 x  4 if
 x

if
x5
x3
Sketch graphs of the following piecewise functions.
1  x 2

f (x)   x  4

 x
if
x  1
if  1  x  2
if
x2

7  x

g ( x)  3
 1

x7
if
x0
if
0 x2
if
x4
• A very important piecewise function is f(x)=|x|. Although
it doesn’t appear to be a piecewise function, it is.
x
f ( x)  x  
 x
if
x0
if
x0
• The domain of this function is all real numbers and the
range is [0, ∞).
• Consider the function f ( x )  x  2
• Evaluate:
f (  4)
f (  2)
f (0)
• Sketch a graph of the function
using transformations.
• To write the function f ( x )  x  2 as a piecewise-defined
function, use the same general principles used for the
basic absolute value function.
if
 x  2
f (x)  x  2  
 ( x  2) if

x20
x20
which simplifies to:
x  2
f ( x)  
 x  2
if
x  2
if
x  2
• Notice how these piecewise lines correspond with the
graph you’ve drawn using transformations.
Consider the function f ( x )  3 2 x  1  6
• Find the x and y-intercepts.
• Write f(x) as a piecewise-defined function and sketch a
graph.


f (x)  


Consider the function f ( x ) 
2 x 1
x 1
• Write f(x) as a piecewise-defined function and sketch a
graph.


f (x)  


Write the following function as a piecewise-defined function.
[Hint: There will be three cases to consider.]
f ( x)  2 x  1  x  4




f (x)  



• Sketch a graph of the function f ( x )  x 2  9 by first
sketching a graph of x 2  9 and then reflecting the
negative portion(s) of the graph across the x-axis.
6.2 – Operations on Functions
• Given functions f(x) and g(x), we can create new
functions by adding, subtracting, multiplying, and dividing
f and g.
( f  g )( x )  f ( x )  g ( x ) 

( f  g )( x )  f ( x )  g ( x ) 

( fg )( x )  f ( x ) g ( x )

 f 
f ( x)
(
x
)

 
g ( x)
 g 
Domain: All values of x for which
both f(x) and g(x) are defined.
Domain: All values of x for which both
f(x) and g(x) are defined, but g(x)≠0.
Consider the functions f ( x ) 
x  4 and g ( x ) 
1
x2
Find the following functions and their domains.
( g  f )( x )
( fg )( x )
 g 
  (x)
 f 
 f 
  (x)
 g 
.
Function Composition
• Two functions f(x) and g(x) can also be combined using
function composition.
(f
g )( x )  f ( g ( x ))
• The domain of this composition is all values of x for which
both g(x) AND f(g(x)) are defined.
• Given f(x) and g(x), evaluate the indicated compositions.
f ( x )  x  7,
2
(f
g )(1) 
(g
f )(2) 
g ( x)  3  x
Use the graph below to evaluate the indicated compositions.
(g
f )(2)
(g
g )(1)
Given f(x) and g(x) below, find the indicated functions and
their domains.
f ( x )  x  5 x  1,
2
(f
g )( x )
(g
f )( x )
( h h )( x )
g ( x )  3 x  2,
h( x)  x  4
2
Given f(x) and g(x) below, find the indicated functions and
their domains.
f ( x) 
x  3,
(g
f )( x )
(f
g )( x )
(h
f )( x )
g ( x )  2 x  7,
h( x)  x  1
2
f ( x) 
1
x2
(f
g )( x )
(f
h )( x )
(g h
,
f )( x )
g ( x )  x  7; h ( x ) 
2
1
x
3( x  1)  1
2
• Find functions f(x) and g(x) such that f ( g ( x )) 
• Given f(x) and f(g(x)) below, what is g(x)?
f ( x )  3 x  7;
f ( g ( x ))  6 x  4
2
x 1
2
6.3 – One-to-One Functions and Inverses
• A function is said to be one-to-one if every x-value in the
domain corresponds to a different y-value in the range.
No two x-values can have the same function value.
• Mathematically, a function f is one-to-one if whenever
f ( x1 )  f ( x 2 ) , it must be that x1  x 2 .
• Graphically, a function is one-to-one if it passes the
Horizontal Line Test:
A function is one-to-one if and only if no horizontal line
intersects the graph at more than one point.
Determine whether the following graphs represent one-toone functions or not.
Determine whether the following functions are one-to-one. If
not, how can you restrict the domain to make it a one-toone function?
f (x)  x  7
g (x) 
x3
h(x)  x  9
2
f (x) 
g ( x) 
1
( x  2)
1
x2
2
Inverses
• If f ( x ) is a one-to-one function, then it has an inverse
1
function f ( x ) defined by:
f
1
( y)  x 
f ( x)  y
where y is any value in the range of f.
• If f has domain A and range B, then its inverse function
has domain B and range A.
• Graphically, the graph of f ( x ) is a reflection of the graph
of f ( x ) across the line y=x.
1
• Suppose that f is a one-to-one function with the following
function values:
f (1)  5, f (2)  7, f (  1)  3, f (3)  2
Determine the following inverse function values.
f
f
1
1
(2)
(3)
• Consider the graph of the one-to-one function f below.
Domain of f:____________
Range of f:_____________
• Evaluate:
f
f
f
1
(  1) 
1
(  2) 
1
(2) 
• Sketch a graph of the inverse
function on the same set of axes.
• Domain of
• Range of
f
f
1
1
:__________
:___________
By only sketching a graph of the function f below, determine
the domain and range of the inverse of f.
f ( x)  
1
2
x
1
• Domain of f :___________
• Range of f  1 :____________
•
If two functions are inverses of each other, then they satisfy the
following properties:
1
f ( f ( x ))  x
f
1
( f ( x ))  x
•
In other words, a function and its inverse “undo” each other.
•
Verify that the following functions are inverses of each other.
f ( x) 
1
3 x
; g ( x)  3 
1
x
• To find the inverse of a one-to-one function, write y=f(x),
interchange the variables x and y, and solve for y. The
function attained is the inverse function.
• Find the inverses of the following functions.
f ( x )  ( 3 x  1)  1
5
f ( x) 
2x 1
3x  7
Find the inverse for the function below and sketch a graph of
both the function and its inverse. State the domain and
range of each.
f ( x) 
x35
• Domain of f:____________
• Range of f:_____________
• Domain of
• Range of
f
f
1
1
:__________
:___________
Things to Do Before Next Meeting:
• Work on Sections 6.1-6.3 until you get all green
bars!
• Write down any questions you have.
• Continue working on mastering 5.4-5.6. After
you have all green bars on 5.1-5.6, retake the
Chapter 5 Test until you obtain at least 80%.
• Make sure you have taken the Chapter 7 Test
before our next meeting.