Section 6 – 3:
Operations on Functions
The 4 basic operations that we perform on two functions are addition, subtraction, multiplication and
division. The notation for each operation is written as a single function. It may be best to rewrite (or at
least rethink) the notation in a form that you are more familiar with.
The notation for adding two functions
f (x) and g(x)
is
(f
The notation for subtracting two functions
f (x) and g(x)
+ g)(x)
is
and means
f (x) + g(x)
The notation for multiplying two functions
f (x) and g(x)
is
− g)(x)
and means
f (x) − g(x)
The notation for dividing two functions
f (x) and g(x)
⎛ f⎞
is ⎜ ⎟ (x)
⎝ g⎠
( f g)(x)
and means
and means
f (x)
g(x)
f (x) • g(x)
Math 120 Section 6 – 3
(f
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© 2012 Eitel
Adding two functions f (x) and g(x)
The notation for adding two functions f (x) and g(x) is
(f
+ g)(x) and means f (x) + g(x)
Finding an algebraic expression for ( f + g)(x)
Example 1
Example 2
Find ( f + g)(x) if
Find ( f + g)(x) if
f (x) = 7x − 4 and g(x) = −5 x + 1
(f
f (x) = x 2 − 3x + 1 and g(x) = −4 x 2 + 5x − 8
(f + g)(x)
+ g)(x) means to add f (x) and g(x)
means to add f(x) and g(x)
f (x) + g(x)
f (x) + g(x)
f (x) + g(x) = (x 2 − 3x + 1) + (−4 x 2 + 5x − 8)
f (x) + g(x) = (7x − 4) + (−5x + 1)
= 7x − 4 − 5x + 1
= 2x − 3
(f
= x 2 − 3x + 1− 4x 2 + 5x − 8
= −3x 2 + 2x − 7
+ g)(x) = 2x − 3
(f + g)(x) =
− 3x 2 + 2x − 7
What is the difference between ( f + g)(x) and (g + f )(x)
You can be asked to find ( f + g)(x)
You can be asked to find (g + f )(x)
which means f (x) + g(x)
which means g(x) + f (x)
Example 3
Example 4
Find ( f + g)(x) if
f (x) = 2x + 3 and g(x) = 5x − 7
Find (g + f )(x) if
g(x) = 5x − 7 and f (x) = 2x + 3
(f
(g + f )(x)
+ g)(x) means to add f (x) and g(x)
f (x) + g(x)
means to add g(x) and f (x)
g(x) + f (x)
f (x) + g(x) = (2x + 3) + (5x − 7)
= 2x + 3+ 5x − 7
= 7x − 4
g(x) + f (x) = (5x − 7) + (2x + 3)
= 5x − 7 + 2x + 3
= 7x − 4
(f
(g + f )(x) =
+ g)(x) = 7x − 4
Note:
(f
7x − 4
+ g)(x) will always equal (g + f )(x) because addition is commutative. (f + g)(x) = (g + f )(x)
Math 120 Section 6 – 3
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© 2012 Eitel
Subtracting two functions f (x) and g(x)
The notation for subtracting two functions f (x) and g(x) is
(f
− g)(x) and means f (x) − g(x)
Finding an algebraic expression for ( f − g)(x)
Example 5
Example 6
Find (f − g)(x) if
Find (f − g)(x) if
f (x) = 5x + 1 and g(x) = 2x − 9
(f
f (x) = x 2 − 3x + 1 and g(x) = −4 x 2 + 5x − 8
(f − g)(x)
− g)(x) means f (x) − g(x)
f (x) − g(x) = (x 2 − 3x + 1) − (−4 x 2 + 5x − 8)
distribute the −1
f (x) − g(x) = (5x + 1) − (2x − 9)
distribute the −1
= 5x + 1− 2x + 9
= 3x + 10
(f
means f (x) − g(x)
= x 2 − 3x + 1+ 4 x 2 − 5x + 8
= 5x 2 − 8x + 8
− g)(x) = 3x −10
(f − g)(x) =
5x 2 − 8x + 8
What is the difference between ( f − g)(x) and (g − f )(x)
You can be asked to find ( f − g)(x)
You can be asked to find (g − f )(x)
which means f (x) − g(x)
which means g(x) − f (x)
Example 7
Example 8
Find (f − g)(x) if
f (x) = 2x + 6 and g(x) = 4x + 3
(f
Find (g − f )(x) if
f (x) = 2x + 6 and g(x) = 4x + 3
− g)(x) means f (x) − g(x)
(g − f )(x)
f (x) − g(x) = (2x + 6) − (4 x + 3)
distribute the −1
= 2x + 6 − 4 x − 3
= −2x + 3
(f
g(x) − f (x) = (4 x + 3) − (2x + 6)
distribute the −1
= 4x + 3− 2x − 6
= 2x − 3
− g)(x) = − 2x + 3
Note:
(f
means g( x) − f (x)
(g − f )(x) =
2x − 3
− g)(x) will NOT equal (g − f )(x) unless f (x) = g(x) because subtraction is not
commutative. However ( f − g)(x) = − ( g − f )(x)
Math 120 Section 6 – 3
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© 2012 Eitel
Multiplying two functions f (x) and g(x)
The notation for multiplying two functions f (x) and g(x) is
( f g)(x) and means
f (x) • g(x)
Finding an algebraic expression for ( f g)(x)
Example 9
Example 10
Find ( f g)(x) if
Find ( f g)(x) if
f (x) = 4x and g(x) = x + 2
( f g)(x)
f (x) = x + 2and g(x) = (3x − 5)
( f g)(x)
means f (x) • g(x)
f (x) • g(x) = (4 x)(x + 2)
simplify
f (x) • g(x) = (x + 2)(3x − 5)
simplify
= 4x 2 + 8x
= 3x 2 − 5x + 6x −10
( f g)(x) = 4 x 2 + 8x
Note:
( f g)(x)
means f (x) • g(x)
( f g)(x) = 3x 2 + x − 10
will always equal (g f )(x) because multiplication is commutative. (f g)(x) = (gf )(x)
Math 120 Section 6 – 3
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© 2012 Eitel
Dividing two functions f (x) and g(x)
The notation for dividing two functions f (x) and g(x) is
⎛ f⎞
f (x)
⎜ ⎟ (x) and means
⎝ g⎠
g(x)
⎛ f⎞
Finding an algebraic expression for ⎜ ⎟ (x)
⎝ g⎠
Example 11
Example 12
⎛ f⎞
Find ⎜ ⎟ (x) if
⎝ g⎠
⎛ f⎞
Find ⎜ ⎟ (x) if
⎝ g⎠
f (x) = 4x + 8 and g(x) = 3x + 6
f (x) = x + 3and g(x) = x 2 − 9
⎛ f⎞
f (x)
⎜ ⎟ (x) means
⎝ g⎠
g(x)
⎛ f⎞
f (x)
⎜ ⎟ (x) means
⎝ g⎠
g(x)
f (x) 4x + 8
=
g(x) 3x + 6
factor the expression
4(x + 2)
=
3(x + 2)
Reduce
f (x)
x +3
= 2
g(x) x − 9
factor the expression
x+3
=
(x + 3)(x − 3)
Reduce
f (x) 4
=
g(x) 3
f (x)
1
=
g(x) x − 3
Note:
⎛f⎞
⎛g⎞
⎜ ⎟(x) may NOT equal ⎜ ⎟(x) because division is not commutative.
⎝g⎠
⎝f⎠
f (x) ÷ g(x) ≠ g(x) ÷ f (x)
Math 120 Section 6 – 3
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© 2012 Eitel
Finding the value of ( f + g)(3)
The expression ( f + g)(3) can be evaluated in two different ways.
Example 13A shows that ( f + g)(3) can be found by finding f (3) and g(3) separately and then
adding the separate values
(f
+ g)(3) = f (3) + g(3)
Example 13B shows that ( f + g)(3) can be found by finding the algebraic expression for
(f
+ g)(x)
first and and then putting x = 3 into that expression.
Many students like the technique used in example 13A but there are times when the technique used
in example 5B may be easier for some students.
Example 13A
Example 13B
Find ( f + g)(3) if
Find ( f + g)(3) if
f (x) = 2x + 3 and g(x) = 5x − 7
f (x) = 2x + 3and g(x) = 5x − 7
(f
(f
+ g)(3) can mean to find f (3) and g(3)
to find ( f + g)(x) = f (x) + g(x) first
and then add the values
(f
and then find ( f + g)(3)
+ g)(3) = f (3) + g(3)
by putting 3 in for x into ( f + g)(x)
( f )(3) = 2(3) + 3
(f
f (3) = 9
(f
+ g)(3) = f (3) + g(3)
by putting 3 into ( f + g)(x) in place of x
+ g)(3) = 17
(f
(f
(f
Math 120 Section 6 – 3
+ g)(x) = 7x − 4
Use ( f + g)(x) = 7x − 4 to find ( f + g)(3)
=9+8
(f
+ g)(x) = f (x) + g(x)
= 2x + 3 + (5x − 7)
= 2x + 3+ 5x − 7
g(3) = 5(3) − 7
g(3) = 8
(f
+ g)(3) can mean
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+ g)(3) = 7(3) − 4
+ g)(3) = 21− 4
+ g)(3) = 17
© 2012 Eitel
Finding the value of ( f g)(3)
The expression ( f g)(3) can be evaluated in two different ways.
Example 14A shows that ( f g)(3) can be found by finding f (3) and g(3) separately and then
multiplying the separate values
( f g)(3) =
f (3) • g(3)
Example 14B shows that ( f g)(3) can be found by finding the algebraic expression for
( f g)(x)first
and and then putting x = 3 into that expression.
Many students like the technique used in example 14A but there are times when the technique used
in example 10B may be easier for some students.
Example 14A
Example 14B
Find ( f g)(3) if
Find ( f g)(3) if
f (x) = 4 x and g(x) = x + 2
f (x) = 4 x and g(x) = x + 2
( f g)(3))
( f g)(3) can mean to find
( f g)(x) = f (x)• g(x) first
and then find ( f g)(3) by putting
3 in for x in ( f g)(x)
can mean to find f (3) and g(3)
and then multiply the values
( f g)(3) =
f (3) • g(3)
f (3) = 4 x
f (3) = 4(3) = 12
( f g)(x) = f (x)• g(x)
( f g)(x) = (4x)(x + 2)
( f g)(x) = 4x 2 + 8x
g(x) = x + 2
g(3) = 3 + 2 = 5
( f g)(3) =
If ( f g)(x) = 4x 2 + 8x then
f (3)• g(3)
= 12• 5
( f g)(3) = 4(3) 2 + 8(3)
( f g)(3) = 4(9) + 24
( f g)(3) = 36 + 24
( f g)(3) = 60
( f g)(3) = 60
Math 120 Section 6 – 3
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© 2012 Eitel
⎛ f⎞
Finding the value of ⎜ ⎟ (3)
⎝ g⎠
⎛ f⎞
The expression ⎜ ⎟ (3) can be evaluated in two different ways.
⎝ g⎠
⎛ f⎞
Example 15A shows that ⎜ ⎟ (3) can be found by finding f (3) and g(3) separately and then
⎝ g⎠
⎛ f⎞
f (3)
dividing the separate values ⎜ ⎟ (3) =
⎝ g⎠
g(3)
⎛ f⎞
⎛ f⎞
Example 15B shows that ⎜ ⎟ (3) can be found by finding the algebraic expression for ⎜ ⎟ ( x ) first
⎝ g⎠
⎝ g⎠
and and then putting x = 3 into that expression.
Many students like the technique used in example 15A but there are times when the technique used
in example 15B may be easier for some students.
Example 15A
Example 15B
⎛ f⎞
Find ⎜ ⎟ (3) if
⎝ g⎠
f (x) = 4x and g(x) = x + 2
⎛ f⎞
Find ⎜ ⎟ (3) if
⎝ g⎠
f (x) = 4x and g(x) = x + 2
⎛ f⎞
⎜ ⎟ (3) can mean to find f (3) and g(3)
⎝ g⎠
first and then divide the values
⎛ f⎞
f (3)
⎜ ⎟ (3) =
⎝ g⎠
g(3)
⎛ f⎞
⎜ ⎟ (3) can mean
⎝ g⎠
⎛ f⎞
to find ⎜ ⎟ (x) first and then find
⎝ g⎠
⎛ f⎞
⎛ f⎞
⎜ ⎟ (x) by putting 3 in for x in ⎜ ⎟ (x)
⎝ g⎠
⎝ g⎠
f (x) = 4x
f (3) = 12
and
g(x) = x + 2
g(3) = 5
⎛ f⎞
4x
⎜ ⎟ (x) =
⎝ g⎠
x+2
so
⎛ f⎞
4(3)
⎜ ⎟ (3) =
⎝ g⎠
3+2
so
⎛ f⎞
f (3) 12
=
⎜ ⎟ (3) =
⎝ g⎠
g(3)
5
Math 120 Section 6 – 3
⎛ f⎞
12
⎜ ⎟ (3) =
⎝ g⎠
5
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© 2012 Eitel
Finding an algebraic expression for f (g(x)) and g( f (x))
Example 16
Example 17
If f (x) = −2 x + 1 and g(x) = 3x
If f (x) = −2 x + 1 and g(x) = 3x
f ( g( x )) means to put the expression for
g( f ( x ))means to put the expression for
g(x) into f (x) in place of x
f (x) into g( x) in place of x
f ( g( x )) means f (3x )
g( f ( x )) means g(−2x + 1)
If f (x) = −2 x + 1 and g(x) = 3x
then
If g(x) = 3x and f (x) = −2 x + 1
then
f ( g( x )) = f (3x ) = −2(3x) + 1
g( f ( x )) = g(−2x + 1) = 3(−2 x + 1)
then find f (g( x ))
then find g( f ( x ))
f ( g( x )) = −6 x + 1
g( f ( x )) = −6 x + 3
Note: f (g(x)) may NOT equal g( f (x))
Example 18
Example 19
If f (x) = −2 x + 1 and g(x) = −x + 4
If f (x) = −2 x + 1 and g(x) = −x + 4
then find f (g( x ))
then find g( f ( x ))
f ( g( x )) means to put the expression for
g( f ( x )) means to put the expression for
g(x) into f (x) in place of x
f (x) into g( x) in place of x
f ( g( x )) means f (−x + 4 )
g( f ( x )) means g(−2x + 1)
If f (x) = −2 x + 1 and g(x) = −x + 4
then
If f (x) = −2 x + 1 and g(x) = −x + 4
then
f ( g( x )) = f (−x + 4 )
g( f ( x )) = g(−2x + 1)
= −2(−x + 4) + 1
= 2x − 8 + 1
= −(−2x + 1) + 4
= 2x − 1+ 4
f ( g( x )) = 2x − 7
Math 120 Section 6 – 3
g( f ( x )) = 2x + 3
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© 2012 Eitel
Example 20
Example 21
If f (x) = −2 x + 1 then find f ( f ( x ))
If
f ( f ( x )) = f (−2x + 1)
g( g( x )) = g(4x + 5)
f (−2 x + 1) = −2(−2x + 1) + 1
g( 4x + 5) = 4(4 x + 5) + 5
= 4x − 2 + 1
= 16x + 20 + 5
f ( f ( x )) = 4x + 1
Math 120 Section 6 – 3
g(x) = 4x + 5 then find g( g( x ))
g( g( x )) = 16x + 25
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© 2012 Eitel