Download (f g) (x) - bcarroll01

x
52
63x
15
f(x) = 3x
1.
Addition
2.
Subtraction
3.
Multiplication
4.
Division
If you are given two functions:
f(x) and g(x)
Their sum is:
(f + g) (x) = f(x) + g(x)

Example:
f(x) = 5x + 4
g(x) = 8x – 2
(f + g) (x) = f(x) + g(x)
5x + 4 + 8x – 2
= 13x + 2

1.
2.
3.
Find (f + g) (x) if:
f(x) = 4x² + 3x + 2
g(x) = 2x² - 5x – 6
(f + g) (x) = f(x) + g(x)
= 4x² + 3x + 2 + 2x² - 5x – 6
= 6x² -2x-4

1.
2.
3.
Find (f + g) (x) if:
f(x) = 5x² + 4x – 2
g(x) = 2x² - 3x + 2
(f + g) (x) = f(x) + g(x)
= 5x² + 4x – 2 + 2x² - 3x + 2
=7x²+ x
1.
Addition
2.
Subtraction
3.
Multiplication
4.
Division


Just as we could add to functions, we can also
subtract two functions.
(f – g) (x) = f(x) – g(x)

1.
Example: Find the difference (f – g) (x) if:
f(x) = 8x + 4
g(x) = 7x – 2
(f – g) (x) = f(x) – g(x)
8x + 4 –( 7x – 2 )
*For subtraction, use parentheses
=x+6

1.
2.
3.
4.
Find (f – g) (x) if:
f(x) = 8x² + 4x – 2 and g(x) = 3x² - 2x + 1
(f – g) (x) = f(x) – g(x)
= 8x² + 4x – 2 – (3x² - 2x + 1)
= 8x² + 4x – 2 – 3x² + 2x - 1
= 5x²+6x-3

1.
2.
3.
4.
Find (f - g) (x) if:
f(x) = 5x² + 4x – 2
g(x) = 2x² - 3x + 2
(f - g) (x) = f(x) - g(x)
= 5x² + 4x – 2 – (2x² - 3x + 2)
= 5x² + 4x – 2 – 2x² + 3x - 2
= 3x²+7x -4
1.
Addition
2.
Subtraction
3.
Multiplication
4. Division

As with addition and subtraction, we can
also multiply two functions.
For the two functions f(x) and g(x), the we
find the product by:
(f • g) (x) = f(x) g(x)

Let f(x) = 8x
and
g(x) = 2x²
Find (f • g) (x):
1) (f • g) (x) = f(x) • g(x)
= 8x • 2x²
= 16x³
f(x) = 3x
g(x) = 2x – 4
Distribute the 3x over the 2x and -4
Find (f • g) (x):
= 6x² - 12x
1) (f • g) (x) = f(x) • g(x)
2)
= 3x (2x – 4)
Here, we must use our Distributive Property
f(x) = 3x
Find:
g(x) = x + 2
a. (f • g) (x) = 3x² + 6x
b. (g • h) (x) = 2x² + 3x - 2
c. (f • h) (x) = 6x² - 3x
h(x) = 2x - 1
1.
Addition
2.
Subtraction
3.
Multiplication
4. Division

Division of functions works just as the
other 3 operations:
f(x)
(f ÷ g) (x) =
g(x)

Ex:
f(x) = 2x² + 3x – 4
and
f(x)
(f ÷ g) (x) =
g(x)
2x² + 3x – 4
=
5x² - 2x - 1
g(x) = 5x² - 2x - 1

Ex:
f(x) = x² - x – 12
and
g(x) = x² - 2x - 8
f(x)
(f ÷ g) (x) =
g(x)
x² - x – 12 (x + 3)(x - 4)
(x + 3)
=
=
=
x² - 2x - 8
(x +2 ) (x – 4) (x +2 )
1.
Addition
2.
Subtraction
3.
Multiplication
4. Division

There is one more operation that can be
performed on functions
 Composition
of Functions
(f ○ g) (x) = f ( g (x) )
Pronounced “F of G of X”
f (x) = 4x and
Find (f ○ g) (x)
g(x) = x + 2
1. (f ○ g) (x) = f ( g(x) )
2.
=f(
)
This leaves us with: f (x + 2)
We need to find f(x + 2)
In the f(x) function, replace the x with x + 2
f(x) = 4x
(x + 2)
= 4x + 8
Therefore, f (g (x)) = 4x + 8
(f ○ g) (x) = f ( g (x) )
= f (2x – 4)
= 3 x(2x – 4)
= 6x - 12
(f ○ g) (x) = f ( g (x) )
= f (8x)
x +4
= (8x)
= 8x + 4
(g
(f ○ f)
g)(x)
(x)==gf ( fg(x)
(x)))
=g
f (x²
(x +
- 2x)
5)
= (x
(x²+- 5)²
2x)-+2(x+5)
5
= (x
x² -+2x
5)²+- 52x – 10
(f ○○g)
(g
f) (x) = g
f ((gf (x) )
=g
f (x
(2x)
+ 2)
= 2x
2 (x++2 2)
= 2x + 4
Find (g
(f ○○g)
f )(x)
(x)==fg(g( f(x)
(x)) )
=g
f (x
(2x²)
– 5)
= 2x²
2 (x-–55)²
1.
f(x) = x² ; g(x) = x + 4
= x² + 4
2.
f(x) = 4x ; g(x) = x – 3
= 4x - 12
3.
f(x) = 2x ; g(x) = x + 2
= 2x + 4
4.
f(x) = x² ; g(x) = 2x
= 4x²
5.
f(x) = x + 1 ; g(x) = 3x
= 3x + 1
We can use these same formulas to
calculate problems substituting numbers
in for x.
f(3) +(fg(3)
For example:=Find
+ g) (3) for the two
functions: = 5(3) + 2 + 2(3) - 2
2 + 6=- 2x
2 -2
f(x) = 5x + 2 =15
and+ g(x)
= 21
Find: a. (f – g) (4) = 18
= f(4) – g(4)
b. =(f8•(4)
g) +
(1)2 - 4(4)
= 40
= f(1)
32 +•2g(1)
- 16
(8(1)
+ 2)• 4(1)
c. =(f18
÷ g) (2)
= (10)
f(2) (4)
18 9
= g(2)
= 8 =
= 40
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