# Advanced Functions and Equations Part

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```					Advanced Functions and
Equations
Part I
Composition and operations on functions,
Transformations
Sections 3.4, 3.5
Equations
Composition and operations on functions
Section 3.5
Operations of functions
expl:
A certain company has two factories. The first factory has a
revenue of R1(x) = 3x + 40 where x is the number of
units made. The second factory has a revenue of
R2(x) = 4x + 50.

1.) What is the total revenue of both factories?
2.) How much more does factory 2 make?
For part 1, you need to find the sum of the revenues.
Algebraically, let’s say
R1 ( x)  R2 ( x)
 3x  40  4 x  50
 7 x  90
For part 2, you need to find the difference of the revenues.
Algebraically, let’s say
R2 ( x)  R1 ( x)                         Put parentheses
around 3x + 40

 4 x  50  3x  40                     because you’re
subtracting.
 4 x  50  3x  40
 x  10
Notation
The following notation is often used.

( f  g )( x)  f ( x)  g ( x)
( f  g )( x)  f ( x)  g ( x)
( f  g )( x)  f ( x)  g ( x)
f        f ( x)
 ( x ) 
g               , g ( x)  0
         g ( x)
Domains: all real numbers in both domains of f and g
and, in the case of (f/g)(x), exclude those numbers that
make g(x) = 0
expl: #5, pg 306
Let    f ( x)  x
g ( x)  3x  5          Remember the
domain will be those
Find    a.) f + g            numbers that satisfy
b.) f – g               both functions.
In the case of the
c.) fg               quotient, we exclude
d.) f / g            those x’s that would
and their domains.     cause division by
zero.
a.)   f ( x)  g ( x)         b.)   f ( x)  g ( x)
 x  3x  5                   x  (3 x  5)
 x  3x  5
domain:     x0               domain:    x0

c.)   f ( x)  g ( x)     d.)    f ( x)     x

 x 3x  5               g ( x) 3 x  5
5
domain:       x  0, x 
domain:    x0                                     3
Composition
expl:
A popular umbrella manufacturer has this information.
P(x) = .0025x2 + .2x + 4, where P(x) represents the
monthly profit of the company and x represents the
number of umbrellas sold that month, x  0
U(t) = 80t + 20, where U(t) represents the number of
umbrellas sold per month and t represents the number of
days it rains that month, 0  t  31
We are interested in how profit is related to the number of
days of rain. Let’s find the equation.
A specific example
Say we have had 12 days of rain this month. Find the
number of umbrellas we can expect to sell and the profit
this should yield.

U (12)  80(12)  20  980               If it rains 12
days, we can
expect to sell
980
…Now find             P(980)          umbrellas…
the profit we
make if we             .0025(980)  .2(980)  4
2

sell 980
umbrellas.             .0025  960400  196  4
 2601 If we sell 980
umbrellas, we will
make a profit of \$2601.
Number of
Number of
umbrellas
days of rain
U(t)

Number of   Profit
umbrellas   P(x)

We’d like to bypass the
number of umbrellas
stage. We’d like a
formula that gives us
profit given number of
days of rain.
P( x)  .0025x 2  .2 x  4
…where x represents the number of umbrellas.
But U(t) also represents the number of umbrellas.
So we can substitute U(t) in for x to find P(U(t)).

Apply the rule of
 P(U (t ))                           P(x) to the number
 P(80t  20)                               80t + 20

 .002580t  20  .280t  20  4
2

                        
 .0025 6400t 2  3200t  400  .280t  20  4
 16t 2  8t  1  16t  4  4
 16t 2  24t  9
So P (t )  16 t  24 t  9 where t is the number of
2

days of rain and P(t) is the profit.

We used composition of functions to find P(U(t)). Now we
know the relationship between profit and number of days
of rain.

Check the specific example. If it rains 12 days, what’s the
profit?
P(12 )  16 (12 )  24 (12 )  9  2601
2

Composition notation:
( f  g )(x)  f ( g ( x))
expl: #14 ac, pg 306
Let   f ( x)  3x  2              Remember f(x) and
g ( x)  2 x  1
2              g(x) are rules. They
tell you what to do
a.) Find    f  g 4 .             to x, no matter
c.) Find    f  f 1 .            what that value is
called.

Verify results (will do part a on grapher).
For part a,
 f  g 4
 f ( g ( 4))                 g(4) = 2(4)2 – 1 = 31

 f (31)
 3(31)  2                      For part c,

 95                                     f  f 1
 f ( f (1))
f(1) = 3(1) + 2 =              f (5)
5
 3(5)  2
 17
Verify part a on grapher
 f  g x                First, find the
 f ( g ( x ))              equation for
y   f  g x 
 f (2 x  1)
2

 3( 2 x  1)  2
2

 6x  3  2
2

 6x2 1

Then graph it and use the calculator to calculate the y
value when x is 4.
On the Graph
screen, press 2nd
TRACE (CALC) to
get to the
Then select Value.
Give it an input of 4
for x, it will tell you
what y is.

In order for the Value or          EVAL on the
EVAL function to work, your         TI-85 or 86
x value must be within the
window dimensions.
Domain of f ( g ( x))
all real numbers x that are in the domain of g and such
that g(x) is in the domain of f.

expl: Find f(g(x)) and determine its domain.
f ( x)  x
g ( x)  3 x  2                      First find
f(g(x)).
f ( g ( x))
 f (3x  2)
 3x  2
The domain of this composite
function are those numbers that
would work in g(x) = 3x + 2, all real
numbers, and those g(x) values
that would work in f. Now f requires
that its inputs are not negative.
So…

Domain:   3x  2  0
2
x
3
You could also think
of this as simply
finding the domain,
those numbers that
work, of the
3x 
y function2

Look at its graph!
2
Domain:   x
3
expl: # 52
Show that f(g(x)) = x and g(f(x)) = x for

f ( x)  4  3 x

g ( x)  4  x 
1                          We will see
later how this
3                        proves the two
functions are
inverses.
Show f(g(x)) = x.                   Show g(f(x)) = x.

Apply the
f ( g ( x))          function’s
g ( f ( x))
1                  rule         g (4  3x)
 f  4  x 
3         
 4  ( 4  3 x) 
1
1                            3
 4  3 4  x 
 4  4  3 x 
3                            1
 4  4  x                         3
Simplify.
 3 x 
1
 44 x         Watch the
order of          3
x
operations!       x
Worksheet
“Functions: Composition and operations” provides practice
in performing operations and compositions as well as
determining domain.

3.5 homework: 1, 9, 13, 27, 32, 33, 47, 51, 55, 65, 67, 77
Equations
Transformations
Section 3.4
Worksheets
“Transformations” introduces several transformations. Most
of the functions we will deal with this semester are
transformations of the functions discussed in “Library of
functions”. These functions, such as y = x 2, y = x, and
others, are called mother functions.

“Part II” of this worksheet summarizes the transformations
for which you will be responsible.

“Transformations 2” and “Library of functions and
transformations 2” provide more practice.
Intro to Transformations worksheet
Part I:
to make sure you are on the right track.
1. Complete the table and graph f ( x)  x and h( x)  x 2  3 .
2

x          -3    -2     -1     0      1      2      3

f ( x)  x 2

h( x )  x  3
2
x         -3   -2   -1   0   1   2   3

f ( x)  x 2    9   4    1    0   1   4   9

h( x)  x 2  3 12    7    4    3   4   7   12

In the table,
each h(x)                                    Notice how
value is 3                                   for each x,
more than the                                   the point is
correspondin                                     moved up
g f(x) value                                   3 units to
because we                                      make h(x).
each f(x).
Transformations worksheet
Let c be a positive    Transformation of     Number of question
real number.              graph             which is example

c  f ( x), c  1       Vertical stretch by
5
factor of c

Vert. compression
c  f ( x), 0  c  1   by factor of c              6

Horizontal shift
f ( x  c)         to right c units            4

Horizontal shift
f ( x  c)         to left c units             3
Transformations worksheet cont.
Let c be a positive   Transformation of   Number of question
real number.             graph           which is example

f ( x)  c         Vertical shift
up c units                1

f ( x)  c         Vertical shift
2
down c units

 f (x)            Reflection about          7
x axis

f (  x)          Reflection about          8
y axis
You can think through transformations by imagining what is
happening to the y values.
For instance, if you compare
y1  x and y 2  5x , you’ll notice the second
2                  2

function’s y values are merely five times the first
function’s y values. So each point is stretched away from
the x-axis. The graph will be elongated. So we say the
graph is vertically stretched by a factor of 5. Check it out
by graphing both on your calculator!

3.4 homework: 1 – 12, 13 – 16, 17, 19, 21, 23, 27, 33, 46,
53, 63, 69

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