To find the sum of two functions f and g, add the corresponding terms of f(x) and g(x). To find the difference, subtract the terms of g(x) from f(x), distributing the negative sign. To find the product, multiply corresponding terms using FOIL. To find the quotient, divide the functions. The composition f o g means to substitute g(x) for each x in f(x). The domain of a composition is numbers where g(x) is in the domain of f.

2. 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) = 2x + 3
g( x) = 4x + 1
2
3
f + g = 2x + 3 + 4x +1
2
3
= 4x + 2x + 4
3
2
Combine like
terms & put in
descending
order

3. 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) = 2x + 3
2
f − g = 2x
2
+ 3 − (4x
g( x) = 4x + 1
3
3
+ 1)
Distribute
negative
2
= 2 x + 3 − 4 x − 1 = −4 x + 2 x + 2
2
3
3

4. 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) = 2x + 3
g( x) = 4x + 1
2
(
3
)(
)
f ⋅ g = 2x + 3 4x +1
2
3
= 8 x + 2 x + 12 x + 3
5
2
3
FOIL
Good idea to put in
descending order
but not required.

5. The quotient f /g
f
f ( x)
 ( x ) =
g
g ( x)
 
To find the quotient of two functions, put the first one
over the second.
f ( x) = 2x + 3
2
f 2x + 3
= 3
g 4x +1
2
g( x) = 4x + 1
3
Nothing more you could do
here. (If you can reduce
these you should).

6. 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.

7. COMPOSITION
OF
FUNCTIONS
“SUBSTITUTING ONE FUNCTION INTO ANOTHER”

8. 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) = 2x + 3
g( x) = 4x + 1
2
(
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

9. ( 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) = 2x + 3
g( x) = 4x + 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

10. ( 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) = 2x + 3
g( x) = 4x + 1
2
(
3
)
2
f  f = 2 2x + 3 + 3
2

11. 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.