Test Instructions

Part 1 (Calculator allowed; Multiple
Choice)
Turn in you scantron and Part 1
 Put your calculator away


Part 2 (NO CALCULATOR; Written)

Use of calculator will be considered
cheating
6.1 Operations on Functions
Review: What is a function?
A relationship where every domain
(x value) has exactly one unique range
(y value).
 Sometimes we talk about a FUNCTION
MACHINE, where a rule is applied to
each input of x

Function Operations
Addition :  f  g ( x)  f x  g x
Multiplica tion :  f  g x  f x g x
Subtractio n :  f  g x  f x  g x
f
f x 
Division :  x  
where gx   0
g x 
g
Function Operations
Adding and Subtracting Functions
Let f  x   3 x  8 and g x   2 x  12.
Find f  g and f - g
 f  g ( x)  f x   g x 
 f  g ( x)  f x   g x 
 (3x  8)  (2 x  12)
 5x  4
 (3x  8)  (2 x  12)
 x  20
When we look at functions we also want to look at
their domains (valid x values). In this case, the
domain is all real numbers.
Multiplying Functions
Let f x   x - 1 and gx   x  1.
Find f  g
2
f x   g ( x)  ( x 2  1)( x  1)
 x3  x 2  x 1
In this case, the domain is
all real numbers because
there are no values that will
make the function invalid.
Dividing Functions
Let f x   x 2 - 1 and gx   x  1.
f 
Find  
g
f x  x 2  1


g x  x  1
( x  1)( x  1)
 x 1
( x  1)
In this case, the domain is
all real numbers EXCEPT
-1, because x=-1 would give
a zero in the denominator.
Let’s Try Some
Let f x   5x 2 - 1 and gx   5x  1.
Find f x  g ( x)
What is the domain?
Find f x   g ( x)
Let’s Try Some
Let f x   5x 2 - 1 and gx   5x  1.
Find f x  g ( x)
What is the domain?
Find f x   g ( x)
Let’s Try Some
Let f x   6 x 2  7x - 5 and gx   2 x  1.
Find f x g ( x)
What is the domain?
Find
f x 
g(x)
Let’s Try Some
Let f x   6 x 2  7x - 5 and gx   2 x  1.
Find f x g ( x)
What is the domain?
Find
f x 
g(x)
Now You Try

Pg 389 #1-2
Composite Function – When you
combine two or more functions

The composition of
function g with
function is written as
g  f x  g  f x
1
1. Evaluate the inner function f(x) first.
2. Then use your answer as the input of
the outer function g(x).
2
Reading Math
The composition (f
g of x.”
o
g)(x) or f(g(x)) is read “f of
Caution!
Be careful not to confuse the notation for
multiplication of functions with composition
fg(x) ≠ f(g(x))
Example – Composition of
Functions
Let f x   x  2 and g x   x 2 . Find g  f  5
Method 1:
Method 2:
g  f x  g  f x
g  f x  g  f x
g x   g ( x  2)  ( x  2) 2
g  f  5   5  2
2
 (7)  49
2
g  f  5  g (5  2)
 g (7)
 (7)  49
2
Let’s try some
Let f x   x3 and g x   x 2  7. Find g  f 2
Solution
Let f x   x3 and g x   x 2  7. Find g  f 2
Solving with a Graphing Calculator
Let f x   x3 and g x   x 2  7. Find g  f 2
Start with the y= list.
Input x3 for Y1 and x2+7 for Y2
Now go back to the home
screen.
Press VARS, YVARS and select
1. You will get the list of
functions.
Using VARS and YVARS enter
the function as Y2(Y1(2).
You should get 71 as a solution.
Real Life Application

You are shopping in a store that is
offering 20% off everything. You also
have a coupon for $5 off any item.
Write functions for the two situations.
Let x = original price.
1.


20% discount: f(x) = x – 0.20x = 0.8x
Cost with the coupon: g(x) = x - 5
You are shopping in a store that is offering 20% off
everything. You also have a coupon for $5 off any
item.
2. Make a composition
of functions:
g  f x   ( g (0.8x))
 0.8 x  5
This represents if
they clerk does the
discount first, then
takes $5 off the
discounted price.
You are shopping in a store that is offering 20% off
everything. You also have a coupon for $5 off any
item.
3. Now try applying the
$5 coupon first, then
taking 20% off:
 f  g x   ( f ( g ( x  5))
 0.8( x  5)
 0.8x - 4
How much more will
it be if the clerk
applies the coupon
BEFORE the
discount?
You are shopping in a store that is offering 20% off
everything. You also have a coupon for $5 off any
item.
4. Subtract the two
functions:
 f  g x   g  f x  
(0.8 x  4)  (0.8 x  5)  1
Any item will be $1
more if the coupon is
applied first. You will
save $1 if you take
the discount, then
use the coupon.