Transcript Algebra 2 compostion of functions

Power Functions
• A power function has the form y  ax b
• -a is a real number
• -b is a rational number
• ( if b is a positive integer then a power
function is a type of polynomial function.)
The sum f + g
 f  g x  f x  g x
This just says that to find the sum of two functions, add
them together. You should simplify by finding like terms.
f x   2 x  3
g x   4 x  1
2
3
f  g  2x  3  4x  1
2
3
 4x  2x  4
3
2
Combine like
terms & put in
descending
order
The difference f - g
 f  g x  f x  g x
To find the difference between two functions, subtract
the first from the second. CAUTION: Make sure you
distribute the – to each term of the second function. You
should simplify by combining like terms.
f x   2 x  3
2

g x   4 x  1
3

f  g  2x  3  4x  1
2
3
Distribute
negative
 2 x  3  4 x  1  4 x  2 x  2
2
3
3
2
The product f • g
 f  g x  f x g x
To find the product of two functions, put parenthesis
around them and multiply each term from the first
function to each term of the second function.
f x   2 x  3
g x   4 x  1
2

3


f  g  2x  3 4x  1
2
3
 8 x  12 x  2 x  3
5
3
2
FOIL
Good idea to put in
descending order.
The quotient f /g
f
f x 
 x  
g x 
g
To find the quotient of two functions, put the first one
over the second.
f x   2 x  3
2
f 2x  3
 3
g 4x 1
2
g x   4 x  1
3
Nothing more you could do
here. (If you can reduce
these you should).
So the first 4 operations on functions are
pretty straight forward.
The rules for the domain of functions would
apply to these combinations of functions as
well. The domain of the sum, difference or
product would be the numbers x in the
domains of both f and g.
For the quotient, you would also need to
exclude any numbers x that would make the
resulting denominator 0.
COMPOSITION
FUNCTIONS
“SUBSTITUTING ONE FUNCTION INTO ANOTHER”
The Composition
Function
 f  g x  f gx
This is read “f composition g” and means to copy the f
function down but where ever you see an x, substitute in
the g function.
f x   2 x  3
2

g x   4 x  1
3

2
f  g  2 4x 1  3
3
FOIL first and
then distribute
the 2
6
3
 32 x  16 x  2  3  32 x  16 x  5
6
3
g  f x  g f x
This is read “g composition f” and means to copy the g
function down but where ever you see an x, substitute in
the f function.
f x   2 x  3
g x   4 x  1
2

3

3
g  f  4 2x  3 1
2
You could multiply
this out but since it’s
to the 3rd power we
won’t
 f  f x  f  f x
This is read “f composition f” and means to copy the f
function down but where ever you see an x, substitute in
the f function. (So sub the function into itself).
f x   2 x  3
g x   4 x  1
2

3

2
f  f  2 2x  3  3
2
Using composition of functions
• A clothing store advertises that it is having
a 25% off sale. For one day only, the store
advertises an additional savings of 10%.
• A. Use a composition of functions to find
the total percent discount.
• B. What would be the sale price of a $40
sweater?
• Let x represent the price.
• f(x)= x - .25x = .75x
• g(x) = x - .10x = .90x
• g(f(x))= .90(.75x)=.675x
• .675(40)=$27
The DOMAIN of the
Composition Function
The domain of f composition g is the set of all numbers x
in the domain of g such that g(x) is in the domain of f.
f g 
1
f x  
x
1
x 1
g x   x  1
The domain of g is x  1
domain of f  g is x x  1
We also have to worry about any “illegals” in this composition
function, specifically dividing by 0. This would mean that x  1 so the
domain of the composition would be combining the two restrictions.
The domain of the composition
function, cont.
• The domain of the new function, after a
function operation, consists of the x values
that are in the domains of both functions.
Additionally, the domain of a quotient does
not include x values that would make the
denominator zero or that would have you
take an even root of a negative number.
• You must pay attention to the order of
functions when they are composed. In
general, f(g(x)) is not equal to g(f(x)).
• (The inner function is substituted into x in
the outer function.)