WARM UP
ACTIVITY:
recall how to multiply a monomial by a
polynomial.
a. 4(3x − 2)
4(3x − 2)
12x -8
Answer
b. x(5x + 3)
x(5x + 3)
2
5xAnswer
+3x
2
c. 3x(x − 3x − 1)
2
3x(x – 3x - 1)
Answer 2
3
3x -9x -3x
2
d. 4x (3x − 5)
2
4x ( 3x - 5)
3
2
Answer
12x -20x
3
3
e. x (x − 2x + 4)
3
3
x (x – 2x + 4)
6
3
4
Answer
x -2x +4x
OPERATION ON
FUNCTIONS
We can multiply and divide functions!
The result is a new function.
Lesson 2
Multiplication and Division
of Functions
MULTIPLICATION
• Product – the product of two
functions and is denoted by
(f●g)(x) ;
• this product is defined as
(𝑓 ●𝑔)(𝑥) = 𝑓(𝑥) ● 𝑔(𝑥).
Example:
Find the product of the functions f(x)=4x+1
and g(x) = x² - 3x + 2.
Solution:
(f⋅g)(x) = f(x)· g(x)
(f⋅g)(x) = (4x+1)(x² - 3x+2)
(f⋅g)(x) = 4x³-12x²+8x + x² - 3x+2
2
(f⋅g)(x) = 4x³-11x +5x+2
Example
Let f(x)=4x-1 and g(x) = x² -3x-2.
Find (f ●g)(x).
Multiply the two given expressions for f(x)
and g(x).
(𝑓 ●𝑔)(𝑥) = 𝑓(𝑥) ● 𝑔(𝑥).
1. Substitute the given functions.
(f●g)(x) = f(x) ● g(x)
(f●g)(x) = (4x-1)(x²-3x-2)
2. Multiply the expressions
(f●g)(x) = (4x-1)(x²-3x-2)
(f●g)(x) = 4x³- 12x² - 8x - x²+3x+2
3. Group similar terms together.
(f●g)(x) = 4x³-12x²-8x-x² + 3x + 2
(f●g)(x) = (4x³) + (-12x² - x²)+(-8x+3x) + (2)
4. Combine similar terms.
(f●g)(x) = (4x³) + (−12x² − x²) + (−8x + 3x) + (2)
(f●g)(x) = 4x3 - 13x2 -5x+2
Thus, (f●g)(x) = 4x³- 13x²-5x+2.
Example
Let f(x)=4x-1 and g(x) = x² -3x-2.
Find (f●g)(0).
Multiply the two given expressions for f(x)
and g(x).
(𝑓 ●𝑔)(𝑥) = 𝑓(𝑥) ● 𝑔(𝑥).
1. Substitute the given functions.
(f●g)(x) = f(x) ● g(x)
(f●g)(x) = (4x-1)(x²-3x-2)
2. Multiply the expressions
(f●g)(x) = (4x-1)(x²-3x-2)
(f●g)(x) = 4x³- 12x² - 8x - x²+3x+2
3. Group similar terms together.
(f●g)(x) = 4x³-12x²-8x-x² + 3x + 2
(f●g)(x) = (4x³) + (-12x² - x²)+(-8x+3x) + (2)
or
(f●g)(x) = 4x³ -12x² - x² -8x + 3x + 2
4. Combine similar terms.
(f●g)(x) = (4x³) + (−12x² − x²) + (−8x + 3x) + (2)
or
(f●g)(x) = 4x³ -12x² - x² -8x + 3x + 2
3
2
(f●g)(x) = 4x - 13x - 5x + 2
Thus, (f●g)(x) = 4x³- 13x²-5x+2.
5. Substitute x = 0 to the
resulting function.
2
(f●g)(x) = 4x³- 13x -5x +2
3
2
(f●g)(0) = 4(0) – 13(0) -5(0) + 2
(f●g)(0) = 0 – 0 - 0 + 2
(f●g)(0) = 2
WARM UP
recall how to divide polynomial
The polynomial we divide with is called
The Dividend
The polynomial we divide by is called
The Divisor
The answer is called
The Quotient
How to fill in as to start?
First, write the dividend inside.
Next, write the divisor outside.
Then, divide and the Quotient will be up.
Quotient
Divisor
Dividend
DIVISION
• Division – the quotient of two
functions and is denoted by ;
(f/g)(x).
• this quotient is defined as
Example:
Let f(x) = x - 1 and g(x) = x² - 3x + 2.
Find(f/g)(x).
Solution:
Divide the two given expressions for f(x)
and g(x).
Example 1:
Let f(x) = x - 1 and g(x) = x² - 3x + 2.
Find(f/g)(x).
1. Substitute the given functions.
(f/g)(x) = f(x) / g(x)
2
(f/g)(x) = x – 1 / x – 3x + 2
Example:
Let f(x) = x - 1 and g(x) = x² - 3x + 2.
Find(f/g)(x).
2. Factor the denominator
(f/g)(x) = __x – 1___
x2 – 3x + 2
(f/g)(x) = ___x – 1___
(x -1 )(x-2)
Example:
Let f(x) = x - 1 and g(x) = x² - 3x + 2.
Find(f/g)(x).
3. Cancel out common factors to simplify.
(f/g)(x) = __x – 1___
x2 – 3x + 2
Thus, (f/g)(x) =
(f/g)(x) = ___x – 1___
(x -1 )(x-2)
1____
x-2
Example 2:
Let f(x) = x3 + 6x2 + 11x + 6 and g(x) = x + 2.
Find(f/g)(x).
Solution:
Divide the two given expressions for f(x) and g(x).
Example 2:
Let f(x) = x3 + 6x2 + 11x + 6 and g(x) = x + 2.
Find(f/g)(x).
1. Substitute the given functions.
(f/g)(x) = f(x) / g(x)
3
2
(f/g)(x) = x + 6x + 11x + 6
x+2
Example 2:
Let f(x) = x3 + 6x2 + 11x + 6 and g(x) = x + 2.
Find(f/g)(x).
2. Divide the two expressions
We can divide the expressions
either factoring or by synthetic
division.
Example 2:
Let f(x) = x3 + 6x2 + 11x + 6 and g(x) = x + 2.
Find(f/g)(x).
-2_|
Multiply
1
1
6
6
-2
11
-8
-6
4
3
0
Since the remainder is 0, the quotient of the
two given functions x2 + 4x + 3
Example 3:
Let f(x) = x - 2 and g(x) = 3x2 -5x - 2.
Find(f/g)(3).
Solution:
Divide the two given expressions for f(x) and g(x).
Example 3:
Let f(x) = x - 2 and g(x) = 3x2 -5x - 2.
Find(f/g)(3).
1. Substitute the given functions.
(f/g)(x) = f(x) / g(x)
(f/g)(x) = __ x – 2___
2
3x -5x - 2
Example 3:
Let f(x) = x - 2 and g(x) = 3x2 -5x - 2.
Find(f/g)(3).
2. Factor the denominator
(f/g)(x) = __ x – 2___
2
3x -5x – 2
(f/g)(x) = __ x – 2___
(x – 2)(3x + 1 )
Example 3:
Let f(x) = x - 2 and g(x) = 3x2 -5x - 2.
Find(f/g)(3).
3. Simplify
(f/g)(x) = __ x – 2___
(x – 2)(3x + 1 )
(f/g)(x) =
1 _
3x + 1
Example 3:
Let f(x) = x - 2 and g(x) = 3x2 -5x - 2.
Find(f/g)(3).
4. Substitute x = 3 to the resulting function.
(f/g)(x) =
1 _
3x + 1
(f/g)(x) =
1 _ Hence, (f/g)(3) = 1/10
3(3) + 1
(f/g)(x) =
1 _= 1
10
9+1