# Study Guide Test 2

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```							              Study Guide Test 2
The following slides present a summary of the concepts
corresponding to Test 2, including various worked out
examples.

The last part of this presentation contains sample
problems similar to the test. These are just a few
problems and do not include all the competencies tested
on Test 2! To be prepared for the test, you must review
the material from all the corresponding sections.

You will find the answers to these sample problems on
the last slide of this study guide.
Section 3.1
Relations and Functions
Objectives
 Understanding the Definitions of
Relations and functions
 Using Function Notation; Evaluating
Functions
 Using the Vertical Line Test
 Determining the Domain of a Function
Given the Equation
Definition of a Relation
   A relation is a correspondence between
two sets A and B such that each element
of set a corresponds to one or more
elements in set B.

Domain: Elements in Set A
Range:   Elements in Set B
Visual of a Relation

Domain                                    Range

24
23
Kenny

Ben

This is a pairing of 3 guys and their ages.
Kenny is 24 years old.
Ben is 23 years old.
Definition of a Function
   A function is a relation such that for each
element in the domain, there is exactly
one corresponding element in the range.
Relation that is a Function
Range
Domain
Output
Input

24
23
Kenny

Ben

This is a pairing of 3 guys and their ages. This relation is a function,
because each input has one and only one output
Kenny is 24 years old.
Ben is 23 years old.
Relation that is not a Function
Domain
(Input)                             Range
Output

24
23
Kenny

Ben

Suppose a bakery is delivering a birthday cake to the 23 year old.
Because there are two 23 year olds there is no way to know for
sure who this cake belongs to. This relation is not a function. A
function has exactly one output value for any input value.
Remember: Only one y-value for each x-value.
Function Notation
 Instead of using the variable y, letters such
as f, g, or h (and others) are commonly
used for function.
 If we want to name a function f, then for
any x-value in the domain, we call the y-
value (or function value) f(x).
 f(x) is read “f of x” or “the value of the
function f at x”
 This is called function notation
Evaluate          f ( x)  x 2  3   for
b) f (a)
a) f (2)
f (a)  a 2  3
f (2)   2   3
2

f ( 2)  4  3
f ( 2)  1
c) f ( x  h)
f ( x  h)                                  f ( x  h)  ( x  h) 2  3

f ( x  h)   x  h   3
2
f ( x  h)  x 2  2 xh  h 2  3

f ( x  h)  x 2  2 xh  h 2  3
Vertical Line Test
A graph in the Cartesian plane is the graph of a function if and
only if no vertical line intersects the graph more than once.
Determine if the graphs are functions.

This graph is a function. No
This graph is not a function.
vertical line intersects the
It does not pass the vertical
graph more than once
line test
Determining the Domain of a
Function Given the Equation.
Domain is the set of all values of x for which the function is defined.
A number x = a is in the domain of a function f if f(a) is a real number.

The domain of every polynomial function is all real numbers.

Definition: Polynomial Function.
n 1
The function f ( x )  an x  an 1 x
n
 an  2 x n  2  ...  a1 x  a0
Is a polynomial function of degree n, when n is a nonnegative integer
and a0 , a1 , a2 ,..., an
are real numbers. The domain of every polynomial function is  , 
Functions with Restricted Domains
Definition: Rational function                          g ( x)
A rational function is a function of the form f ( x)            ,
h( x )

Where g and h are polynomial functions such that       h( x)  0
The domain of a rational function is the set of all real numbers such
that h( x)  0

The domain is all real numbers for which the denominator does
not equal zero. When the denominator is zero, the function is
undefined.
Functions with Restricted Domains
Definition: Root Function

The function f ( x)  n g ( x)    is a root function, where n is an integer
such that n  2

1. If n is even, the domain is the solution to the inequality g ( x)  0
2. If n is odd, the domain is the set of all real numbers for which g(x) is defined.

For even root functions negative values for g(x) are not real numbers
Find the Domain of the Function

f ( x)  3x 3  4 x 2  7 x  2

This is a polynomial function, so the domain of f is all real numbers or
 , 
Find the Domain of the Function
5x
f ( x)  2
x  3x  4

This is a rational function. So the denominator cannot equal zero.

x 2  3x  4  0                 The domain of f is
( x  4)( x  1)  0               x x  1, x  4
x  4  0 or x  1  0           The domain in interval notation is
x  4 or x  1                      ,1  1,4  4, 
Find the Domain of the Function

f ( x)  3x  8

This is a root function where n = 2, an even number. Therefore the
function 3x-8 must be greater than or equal to zero. Any negatives must
be eliminated.

3x  8  0
The domain is all values of be greater than or
3x  8
equal to 8/3.
8
x                                                      8 
3             The domain in interval notation is   3 ,  
      
Find the Domain of the Function
2x  5
f ( x)   5
x7
This is a root function where n = 5, an odd number. Therefore the domain
is all real numbers for which function 2 x  5 is defined.
x7

2x  5
x  7 is a rational function, so all values of x that cause the
denominator to equal to zero must be eliminated. These values
cause the function to be undefined.

The domain of f is
x x  7
The domain in interval notation is
 ,7  7, 
SECTION 3.2
Properties of a Function’s Graph
Objectives
   Determining the Intercepts of a Function
   Determining the Domain and Range of a
Function from its Graph
   Determining Whether a Function is Increasing,
Decreasing, or Constant
   Determining Relative maximum and Relative
Minimum Values of a Function
   Determining Information about a Function from
a Graph
Intercepts: Points on the graph of a function where the
graph either crosses or touches a coordinate axis
•y-intercept: the y-coordinate of the point where the
graph crosses or touches the y-axis
•A function can have one y-intercept or no y-
intercept
•The y-intercept exists if x = 0 is in the domain of
the function
•The y-intercept can be found by evaluating f(0)
•x-intercept: the x-coordinates of the points where
the graph crosses or touches the x-axis
•A function may have several x-intercepts
•The x-intercepts are also called real zeros
•The x-intercepts can be found by finding all real
solutions to the equation f(x) = 0.
Find the x and y intercepts of the function      f ( x)  x 3  2 x 2  x  2

x-intercept Solve f(x) = 0

x3  2 x 2  x  2  0                      y-intercept Evaluate f(0)

x 2 ( x  2)  1( x  2)  0                f (0)  03  2(0) 2  0  2  2
( x  2)(x 2  1)  0
The y-intercept is -2.
x  2  0 x2 1  0
x  2       x2  1
x  1

The x-intercepts are -2, -1 and 1.
Find the Domain and Range of a Function from its Graph

The open dot representing (-1,6)
indicates this point is not included on
the graph. The closed dot representing
(3,10) indicates that this point is included
on the graph. Therefore the domain
includes all x-values between -1 and 3
not including -3.
The domain is (-1,3] .

The range includes all y-values between -1 and
10. The range is [-1,10]
Find the Domain and Range of a Function from its Graph

Notice the graph extends indefinitely
to the right and includes all values of x
greater than or equal to -1. Thus the
domain is [1, )

The horizontal dashed line y = 1, is
called a horizontal asymptote and
indicates that the function will get
arbitrarily close to the line y = -1
as x increases. Thus, the range
include all y values greater than or
equal to -2 but less than -1. [-2.1)
Determining whether a Function is Increasing, Decreasing or Constant

A function f is said to be
•Increasing on an open interval if the graph rises from left
to right
•Decreasing on a open interval if the graph falls from left
to right
•Constant on an open interval if the graph is a horizontal
line on the interval.
Determining whether a Function is Increasing, Decreasing or Constant

The graph is falling,
therefore decreasing
from (-3,1)

The graph is a horizontal
line, therefore constant
from (1,4)

The graph is a rising,
therefore increasing [4, )
Determining Relative Maximum and Relative Minimum Values of a Function

Relative Maximum: Occurs at x = c, when the graph changes
from increasing to decreasing at a point (c, f(c) ). The maximum
is f(c).
Relative Minimum: Occurs at x = c, when the graph changes
from decreasing to increasing at a point (c, f(c) ). The minimum
is f(c).
The graph has a relative maximum at
x = -1 because the graph is changing
from increasing to decreasing at that
point. The maximum is 5.

The graph has a relative minimum at x = 2
because the graph is changing from
decreasing to increasing at that point.
The minimum is -10.
Graphs of Basic Functions; Piecewise
Functions Section 3.3

Objectives
 Sketching the Graphs of the Basic Functions
 Analyzing piecewise Defined Functions
Constant Function
F(x) = b
Linear function with
m = 0. Graph of a
horizontal line.
Identity Function
F(x) = x
Linear function with
m = 1 and b = 0.
Square Function
f ( x)  x 2
Cube Function

f ( x)  x 3
Absolute Value Function

f ( x)  x
Square Root function

f ( x)  x
Cube Root Function

f ( x)  3 x
Reciprocal Function
1
f ( x) 
x
Piecewise-Defined Function
Functions that are defined by rule that has
more than one piece.

2 x  1 if   x 1
f ( x)  
 x  2 if    x 1
2 x  1 if x  1
If f ( x)                   , find f(-2), f(5), f(1)
 x  2 if x  1

f(-2) = 2(-2)-1 = -5
Choose the first piece of the function because -2 < 1

f(5) = 5 + 2 = 7
Choose the second piece of the function because 5 > 1

f(1) = 2(1)-1 = 1
Choose the first piece of the function because 1 = 1
2 x  1 if   x 1
Sketch the piecewise defined function            f ( x)  
 x  2 if    x 1
Graph the first piece
f(x) = 2x - 1 Choose 2 points for the line.
Let x = 1 and any number
smaller than 1
(1,1)           Plot those points placing
(-2,-5)         a solid circle at (1,1)

Graph the second piece
f(x) = x+2
Choose 2 points for the line.
Let x = 1 and any number
greater than 1
(1,3)
(5,7)    Plot those points on the
same graph placing a
hollow circle at (1,3)
Section 3.4
Transformations of Functions

Objectives
   Using Vertical Shifts to Graph Functions
   Using Horizontal Shifts to Graph Functions
   Using Reflections to Graph Functions
   Using Vertical Stretches and Compressions to
Graph Functions
   Using Combinations of Transformations to
Graph functions
Vertical Shifts of Functions

If c is a positive real number
•The graph of y = f(x) + c is obtained by shifting the graph
of y = f(x) vertically upward c units.
•The graph of y = f(x) - c is obtained by shifting the graph
of y = f(x) vertically downward c units.
Horizontal Shifts of Functions

If c is a positive real number
•The graph of y = f(x + c) is obtained by shifting the graph
of y = f(x) horizontally to the left c units.
•The graph of y = f(x – c) is obtained by shifting the graph
of y = f(x) horizontally to the right c units.
Reflections of Functions about the x-Axis

The graph of y = -f(x) is obtained by reflecting the graph of y = f(x)
Using Vertical Stretches and Compressions to Graph Functions

Suppose a is a positive real number:
•The graph of y = af(x) is obtained by multiplying each y-
coordinate of y = f(x) by a.
•If a>1, the graph of y = af(x) is a vertical stretch of y =
f(x)
•If 0<a<1, the graph of y = af(x) is a vertical compression of y =
f(x)
Section 3.5
The Algebra of Functions

Objectives

 Evaluating a Combined Function
 Forming and Evaluating Composite Functions
Algebra of Functions
Let f and g be functions, then for all x such that both f(x) and
g(x) are defined, the sum, difference, product, and quotient of
f and g exist and are defined as follows:

1. The sum of f and g:                      ( f  g )(x)  f ( x)  g ( x)

2. The difference of f and g:               ( f  g )(x)  f ( x)  g ( x)
( fg)(x)  f ( x) g ( x)
3. The product of f and g:
4. The quotient of f and g:                  f
( )( x) 
g
f ( x)
g ( x)
for all g ( x)  0
Examples

f ( x)  2 x  3
3
g ( x) 
x4
h( x )  x  2

 h 
b) ( gh)(6)            c)     (6)
a) ( f  g )(10)                                             f 
( f  g )(10)  f (10)  g (10)   ( gh)(6)  g (6)h(6)    h        h(6)
    (6) 
3                       3                     f        f (6)
2(10)  3                               64
10  4                   64                      62
1     1                                              2(6)  3
17   17
3
(2)  3              4
2     2                       2                      9
Use the graph to evaluate the given expression.

(f+g)(1) =
f(1)+g(1) =
1+3=
4

(g - f)(0) =
g(0)+f(0) =
1-0=
1
Composite Function
Given functions f and g, the composite function, f  g                                             , (also called the
composition of f and g), is defined by
( f  g )(x)  f ( g ( x))
Provided g(x) is in the domain of f.
3
Given f ( x)  2 x  1,         g ( x)         , and h( x)  x  2
x 1

a) ( f g )( x)                               b) ( f     f )( x)                       c ) (h f )(3)
( f g )( x)  f ( g ( x))                    (f      f )( x)  f ( f ( x))          ( h f )(3)  h( f (3)).
3                                      f (2 x  1)                            Since
f(       )
x 1                                                                             f (3)  2(3)  1  7,
2(2 x  1)  1 
3                                                                              h( f (3))  h(7)
2(      ) 1                               4x  2 1 
x 1                                     4x  3
or
6     x 1                                                                        h(7)  7  2  9  3
       
x 1 x 1
x5
x 1
Evaluate Composite functions Using a Graph

Find
( f g )(0)
f ( g (0))
f (1)
1

Find
( g f )(1)
g ( f (1))
g (1)
3
Section 3.6
One-to-One Functions; Inverse Functions

Objectives
 Determining whether a Function is One-to-One
using the Horizontal Line Test
 Understanding and Verifying Inverse functions
 Sketching the Graphs of inverse functions
 Finding the inverse of a one-to-One function
Determine whether each function is one-to-one.

One to one: every                     Not one to one: There
horizontal line intersects            exists a horizontal line that
the graph at most once                intersects the graph more
than once
Inverse function
Let f be a one-to-one function with domain A and range B. Then f -1
is the inverse function of f with domain B and range A.

Composition Cancellation Equations
f ( f 1 ( x))  x For all x in the domain of f -1
f 1 ( f ( x))  x For all x in the domain of f
Use the composition cancellation equations to determine if f and g are
inverse functions.
4                5  3x
f ( x)           and g ( x) 
5  3x                 4

( f g )( x) 
16
Since the composite
f ( g ( x))                                             function does not
20  3(5  3 x)
5  3x                                                equal x; f and g are
f(         )                              16

4
20  15  9 x       not inverse
4
5  3x
                       16
x
functions.
5  3(        )                       35  9 x
4
                
        4        4 
                       
5  3x   4 
 
 5  3(        )
           4    
Find the graph of the inverse of a one-to-one function by interchanging
the coordinates of each ordered pair that lies on the graph of f.
The graph of the inverse is a reflection of the graph of the function
about the line y = x.
Find the inverse of a one-to-one Function
1.   Change f(x) to y
2.   Interchange x and y
3.   Solve for y
4.   Change y to f-1(x)
Write an equation for the inverse function, then state the domain and
range of f and f-1   f ( x)  3 2 x  3

1. y  3 2 x  3
Domain of f is  ,  
2. x  3 2 y  3
Range of f is     , 
3. x  2 y  3
3

x3  3  2 y
x3  3
Domain of f-1 is the Range of f
2
y                  which is  ,  
x3  3
1
4. f ( x)                 Range of f-1is the Domain of f
which is  ,  
2
Sample Problems Test 2

Following you will see some sample questions similar to
those on Test 2.These are just a few problems and do not
include all the competencies tested on Test 2!

To be prepared for the test, you must review the material
from all the corresponding sections.

You can find the answers to these sample problems on
the last slide of this presentation.
1.   A function is a special kind of relation that assigns
to each number in the range exactly one number in the domain.

a. True

b. False
2. The graph below is a graph of a function.

a. True

b. False
3. The domain of f(x) is the set of all real numbers x
for which the function is defined.

a. True

b. False
4. Determine the domain of y = 3x - 1

a. x  1/3
b. x  0
c. x  1/3
d. x = 1/3
5. The graph of the function y = |x – 3| is shown on the grid
to the right.

Find the equation of A.

a.   y = |x + 3| – 1
b.   y = |x – 3| – 1
c.   y = |x + 3| + 1
d.   y = |x – 3| + 1
6. Given f(x) = x2 – 3x, then f(a + b) =

a. a2 + 2ab + b2 – 3a –3b

b. a2 + b2 – 3a –3b

c.   a2 + 2ab + b2 – 3a + b

d. a2 – 4ab + b2
7. f(x) = x3 is a one-to-one function.

a. True

b.   False
8. Find the inverse function of f(x) = 5x – 2.

a. f-1(x)= x + 2
5
b. f-1(x)= x – 2
5
c. f-1(x)= -5x + 2

d. f-1(x)= 5x + 2
9. The function whose graph is shown below is a one-to-one function.

a. True

b.   False
10. If f(x) = 1 and g(x) = x2 + 2, then (g o f) (x) is 1 + 2.
x                                        x²

a. True

b.   False
11. If f(x) = 2x – 1 and g(x) = x + 4, then (f – g) (2) = - 3.

a. True

b.   False
12. Find the x-coordinate of the relative minimum of the
function x3 – 6x – 4 = 0 rounded to four decimal places.

a. -9.6569

b. -9.6325

c. 1.4894

d. 1.4142
Answers for sample questions Test 2:
1. b
2. a
3. a
4. c
5. d
6. a
7. a
8. a
9. b
10. a
11. a
12. d

```
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