College Algebra
Chapter 3
Operations on Functions
and Analyzing Graphs
College Algebra Chapter 3.1 The algebra and composition of functions
Sums and Differences of Functions
For functions f and g with domains of P and Q
respectively, the sum and difference of f and g
are defined by:
 f  g x   f x   g x 
 f  g x   f x   g x 
College Algebra Chapter 3.1 The algebra and composition of functions
Sums and Differences of Functions
f x   x  5 x
find
 f  g x   ?
2
g x   2 x  10
 f  g x   ?
College Algebra Chapter 3.1 The algebra and composition of functions
Sums and Differences of Functions
f x   x  5 x
2
g x  2x 10
if hx   f  g x what would h3  ?
College Algebra Chapter 3.1 The algebra and composition of functions
Products and Quotients of Functions
 f  g x   f x   g x 
f
f x 
  x  
g x 
g
g x   0
College Algebra Chapter 3.1 The algebra and composition of functions
Products and Quotients of Functions
f x   x  2
g x   x  3
find hx    f  g  x 
evaluate h2 and h5
College Algebra Chapter 3.1 The algebra and composition of functions
Products and Quotients of Functions
f  x   x 2  4 x  12 g  x   x 2  7 x  6
f 
find h x     x 
g
state dom ain
College Algebra Chapter 3.1 The algebra and composition of functions
Products and Quotients of Functions
f x   x 2  9 g x   x  1
find H  x    f  g  x 
f
find hx     x 
g
state dom ain
College Algebra Chapter 3.1 The algebra and composition of functions
Composition of Functions
f x   x
2
g x   x  2
hx    f  g x   f g x 
College Algebra Chapter 3.1 The algebra and composition of functions
Composition of Functions
College Algebra Chapter 3.1 The algebra and composition of functions
Composition of Functions
f x   x  2 x  3
g x   x  3
2
find
 f  g x 
f g x  x  3  2x  3  3
2
College Algebra Chapter 3.1 The algebra and composition of functions
Function Decomposition
h x   x  4
h x    f  g  x 
g x   ?
f x   ?
College Algebra Chapter 3.1 The algebra and composition of functions
Function Decomposition
h x    x  1
2
 f  g x   f g x 
f x   ? x 2
g x   ? x  1
College Algebra Chapter 3.1 The algebra and composition of functions
Homework pg 256 1-77
College Algebra Chapter 3.2 one to one and inverse functions
Relations
Functions
One to One Function
If a horizontal line intersects a
graph at only one point, the
function is one to one
College Algebra Chapter 3.2 one to one and inverse functions
Functions
1 to 1 functions
College Algebra Chapter 3.2 one to one and inverse functions
College Algebra Chapter 3.2 one to one and inverse functions
Inverse functions
An inverse function is denoted by
1
This does not mean f  x 
If given coordinates (x,y) the inverse
would have coordinates (y,x)
(3,4) (-2,8) (-7,10)
1


f x
College Algebra Chapter 3.2 one to one and inverse functions
Inverse functions
An inverse must undo operations
taking place in the original equation
College Algebra Chapter 3.2 one to one and inverse functions
Inverse functions
How to find an inverse Algebraically
3
f x   x  4
3
y  x 4
3
x  y 4
3
x4 y
x4  y
3
f
1
x  
3
Use y instead of f(x)
Interchange x and y
Solve for y
The result is the inverse
x4
College Algebra Chapter 3.2 one to one and inverse functions
Inverse functions
 f  f x  x
1
College Algebra Chapter 3.2 one to one and inverse functions
Inverse functions
f x   2 x  1  5
College Algebra Chapter 3.2 one to one and inverse functions
Inverse functions
f x  2 x  14
2
College Algebra Chapter 3.2 one to one and inverse functions
Homework pg 268 1-96
College Algebra Chapter 3.3 Toolbox functions and Transformation
Vertical and Horizontal Shifts
Vertical shift or vertical translation
Given any function whose graph is determined by y
and k>0,
1. The graph of y  f x  k is the graph of f x
shifted upward k units.
2. The graph of y  f x  k is the graph of f x
shifted downward k units.


 f x 


The amount of shift is equal to the constant added to the function
College Algebra Chapter 3.3 Toolbox functions and Transformation
Vertical and Horizontal Shifts
Vertical shift or vertical translation
yx
2
College Algebra Chapter 3.3 Toolbox functions and Transformation
Vertical and Horizontal Shifts
Vertical shift or vertical translation
y  x 4
2
College Algebra Chapter 3.3 Toolbox functions and Transformation
Vertical and Horizontal Shifts
Vertical shift or vertical translation
y  x 2
2
College Algebra Chapter 3.3 Toolbox functions and Transformation
Vertical and Horizontal Shifts
Vertical shift or vertical translation
f x  x
f x   x  1
f x   x  3
College Algebra Chapter 3.3 Toolbox functions and Transformation
Vertical and Horizontal Shifts
Horizontal shift or horizontal translation
Given any function whose graph is determined by y
and h>0,
1. The graph of y  f x  h is the graph of f x
shifted to the left h units.
2. The graph of y  f x  h is the graph of f x
shifted to the right h units.




-Happens when the input values are affected
-Direction of shift is opposite the sign


 f x 
College Algebra Chapter 3.3 Toolbox functions and Transformation
Vertical and Horizontal Shifts
Horizontal shift or horizontal translation
y1  x
2
f x   x
y2  x  2
2
g x   x  3
College Algebra Chapter 3.3 Toolbox functions and Transformation
Vertical and Horizontal Shifts
Horizontal shift or horizontal translation
Graph
f x  x
College Algebra Chapter 3.3 Toolbox functions and Transformation
Vertical and Horizontal Shifts
Horizontal shift or horizontal translation
Graph
f x  x  4
College Algebra Chapter 3.3 Toolbox functions and Transformation
Vertical and Horizontal Shifts
Horizontal shift or horizontal translation
Graph
f x  x  2
College Algebra Chapter 3.3 Toolbox functions and Transformation
Vertical Reflection over x-axis
y1  f x
f x   x  2 x
2
y2   f x

g x    x  2 x
2

College Algebra Chapter 3.3 Toolbox functions and Transformation
Vertical Reflection over x-axis
1
f x   x  2  3
2
1
f x    x  2  3
2
College Algebra Chapter 3.3 Toolbox functions and Transformation
Horizontal Reflections over y-axis
y1  f x
f x   x  8
3
y2  f  x 
g x   x  8
3
College Algebra Chapter 3.3 Toolbox functions and Transformation
Horizontal Reflections over y-axis
f x   x  8
3
g x   x  8
3
College Algebra Chapter 3.3 Toolbox functions and Transformation
y1  f x
f x   x
2
y2  a  f  x 
1 2
g x    x
3
College Algebra Chapter 3.3 Toolbox functions and Transformation
Ways to graph transformations
1) Using a table of values
2) Applying transformations to a parent
graph
a. Apply stretch or compression
b. Reflect result
c. Apply horizontal or vertical shifts
usually applied to a few
characteristic points
College Algebra Chapter 3.3 Toolbox functions and Transformation
f x   x
3
g x   4 x
3
College Algebra Chapter 3.3 Toolbox functions and Transformation
Homework pg 283 1-86
College Algebra Chapter 3.4 Graphing General Quadratic Functions
Shifted Form/Vertex Form
y  ax  h  k
2
Horizontal shift is h units, vertical shift is k units
To put a quadratic equation in shifted form can
be done by completing the square
College Algebra Chapter 3.4 Graphing General Quadratic Functions
Shifted Form/Vertex Form
Completing the square
y  2 x  8x  3
2


y  2x  4x  ____ 3
y  2x  4 x  4 4 3
y   2 x  8x  ____  3
Group variable terms
2
Factor our “a”
2
2


y  2 x  2  4  3
2
y  2x  2  5
2
Add and subtract
then regroup
1

 2 linear coefficient 


Factor trinomial
Distribute and simplify
2
College Algebra Chapter 3.4 Graphing General Quadratic Functions
Shifted Form/Vertex Form
Completing the square
f x   x  4 x  5
2
f x  x  2  9
2
g x   x  5 x  2
2
2
5  17

g x    x   
2
4

College Algebra Chapter 3.4 Graphing General Quadratic Functions
Shifted Form/Vertex Form
Completing the square
f x  3x  5x 1
2
1 2
g x    x  5 x  7
2
College Algebra Chapter 3.4 Graphing General Quadratic Functions
Shifted Form/Vertex Form
Completing the square
Go back 3 pages to find zero’s of each function
Set equation equal to zero and then solve for x
College Algebra Chapter 3.4 Graphing General Quadratic Functions
Shifted Form/Vertex Form
Completing the square
y  2 x  8x  3
2


y  2x  4x  ____ 3
y  2x  4 x  4 4 3
y   2 x  8x  ____  3
Group variable terms
2
Factor our “a”
2
2


y  2 x  2  4  3
2
y  2x  2  5
2
Add and subtract
then regroup
1

 2 linear coefficient 


Factor trinomial
Distribute and simplify
2
College Algebra Chapter 3.4 Graphing General Quadratic Functions
Shifted Form/Vertex Form
Completing the square
y  2x  2  5
2
0  2x  2  5
2
 5  2x  2
5
2
 x  2
2
5

 x2
2
5
2
x
2
2
College Algebra Chapter 3.4 Graphing General Quadratic Functions
Shifted Form/Vertex Form
Completing the square
f x   x  4 x  5
2
f x  x  2  9
2
x  5 or 1
g x   x  5 x  2
2
2
5  17

g x    x   
2
4

College Algebra Chapter 3.4 Graphing General Quadratic Functions
Shifted Form/Vertex Form
Completing the square
f x  3x  5x 1
2
1 2
g x    x  5 x  7
2
College Algebra Chapter 3.4 Graphing General Quadratic Functions
f x  ax2  bx  c
Standard form for a quadratic function has a vertex at
  b   b 
h, k    , f   
 2a  2a  
College Algebra Chapter 3.4 Graphing General Quadratic Functions
Homework pg 295 1-60
College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions
1
Reciprocal Functions
x
Reciprocal Quadratic Functions
1
x2
Asymptotes are not part of the graph, but
can act as guides when graphing
Asymptotes appear as dashed lines guiding
the branches of the graph
College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions
Direction/Approach Notation
As x becomes an infinitely large
negative number, y becomes a
very small negative number
x  0 , y  
x  , y  0
x  , y  0
x  0 , y  
College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions
Horizontal and Vertical asymptotes
a
F x  
k
xh
a
Gx  
k
2
x  h 
The line y=k is a horizontal asymptote if, as x
increases or decreases without bound, y approaches k
The line x=h is a vertical asymptote if, as x approaches
h, |y| increases or decreases without bound
College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions
Horizontal and vertical shifts of rational functions
First apply them to the asymptotes, then calculate
the x- and y-intercepts as usual
College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions
1
g x    2
x
To find x intercept; solve
1
0 2
x
1
x
2
College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions
1
g x  
x2
College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions
1
g x  
1
x2
To find y-intercept; solve
1
g 0  
1
02
To find x-intercept; solve
0
1
1
x2
College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions
College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions
College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions
Homework pg 307 1-56
College Algebra Chapter 3.6 Direct and inverse Variation
College Algebra Chapter 3.6 Direct and inverse Variation
College Algebra Chapter 3.6 Direct and inverse Variation
College Algebra Chapter 3.6 Direct and inverse Variation
Homework pg 321 1-58
College Algebra Chapter 3.7 Piecewise – Defined Functions
Piecewise-Defined Functions
College Algebra Chapter 3.7 Piecewise – Defined Functions
Piecewise-Defined Functions
What is the piecewise function?
 x 2  6 x  3 0  x  6
f x   
x3
3
College Algebra Chapter 3.7 Piecewise – Defined Functions
Piecewise-Defined Functions
College Algebra Chapter 3.7 Piecewise – Defined Functions
Piecewise-Defined Functions
College Algebra Chapter 3.7 Piecewise – Defined Functions
Homework pg 335 1-42
College Algebra Chapter 3 Review
•Composition of functions
•Inverse function
•One-to-one function
•Transformation
•Translation
•Reflection
•Quadratic
•Absolute value
•Linear
•Reciprocal
•Reciprocal quadratic function
•Piecewise-defined functions
•Effective domain
College Algebra Chapter 3 Review
Composition of functions
f x   x 2  5
 f  g x   ?
 f  g x   ?
 f  g x   ?
f 1  x   ?
 f  g x   ?
g x   5 x
Domain and Range?
College Algebra Chapter 3 Review
Toolbox Functions
Know them and their graphs
College Algebra Chapter 3 Review
f x  2x  3  3
3
2
g x  
3
2
 x  2
College Algebra Chapter 3 Review
Variation
The weight of an object on the moon varies directly
with the weight of the object on Earth. A 96-kg object
on Earth would weigh only 16 kg on the moon. How
much would a 250-kg astronaut weigh on the moon?
College Algebra Chapter 3 Review
Piece-Wise Defined Functions
 .5 x  1 x  3

hx    x  5  3  x  5

3 x  5 x  5
.5 x  1
hx    2
x  4x  1
x0
0 x5
College Algebra Chapter 3 Review