W13D3 Function Operations
Warm Up
Lesson 35 Functions Operations
Introduce Lesson with Brainstorm Carousal . Each group will spend 2 minutes at each station. The Paper will be divided
into ”Notes about getting the solution” and ”Questions”. When groups get back to their original station they will look at the
board and summarize all of the notes.
7
2
1
h(x) = −16x 4
g(x) = 5x 3
f(x) = 3x 3
7
2
k(x) = 10x 4
3
m(x) = 81x 7
1
3
n(x) = 3x 4
1
w(x) = 5x 4 − 2x 2
1. Multiplication
f(x) · h(x)
f(x)·g(x)
m(x) · n(x)
2. Division
k(x)
h(x)
m(x)
f (x)
f (x)
g(x)
3. Addition/Subtraction
w(x)+ v(x)
h(x) - n(x)
4. Find the domain
k(x)
n(x)
5. Find the domain
x−3
x−4
6. Find the domain
4
x−9
1
x+4
1
2x + 3
7. Find the domain
f(x)
m(x)
8. Find the domain
1
(x − 2) 6
3
v(x) = 4x 4 + x 2
1
(3x) 4
1
(4x + 5) 2
Solutions
11
1.
15x3
2.
5x 2
−
8
3.
9x 4 − x 2
4.
x≥0
5.
x 6= 0
6.
x 6= −4
7.
IR
8.
x≥2
6
−48x 12
3
1
243x 7
1
27
3x 3
5x2
13
x 28
3
3
1
3x 4 − 16x 4
x≥0
x 6= 0
x 6= 9
x 6= −
IR
x≥0
x≥−
5
2
3
2
RATIONALIZE!
If the denominator of the exponent in the solution OR either original function is even, then the domain is x ≥ 0 (if there is an
expression raised to a power then the whole expression is ≥ 0)
If the denominator of the exponent in the solution AND BOTH original functions are odd, then the domain is IR
If there is a fraction with a variable in the denominator then the domain is x 6= 0 (if there is an expression in the denominator
then the domain is found by setting the Denominator 6= 0)
2
EX 1:
Add then state the Domain and Range
2
2
Must have the same exponent to add like3x 3 and5x 3
So this cannot be added
Both exponent denominators are odd so the domain is IR
1
h(x) = −16x 4
EX 2:
Multiply h(x) · k(x) −160x2
Domain is x ≥ 0
3
Divide
k(x)
5x− 2
=
h(x)
8
Exit Pass Different Denominators
2
m(x) = 81x 7
Multiply m(x) · n(x)
Divide
15
27x 28
x
m(x)
n(x)
3
n(x) = 3x 4
7
= 3x 3 + 5x 3
7
k(x) = 10x 4
Extra Lesson Material - Not today’s Lesson
1. Simplify (3x2 − 7x + 11) − (x3 + 8x2 − x − 4)
2. Simplify 4x3 − 9x2 + 6x + 2 ÷ x2 − 3
1. −x3 − 5x2 − 6x + 15
2.
4x − 9
x −3
2
3
2
4x − 9x + 6x + 2
− 4x3
+ 12x
− 9x2 + 18x + 2
9x2
− 27
18x − 25
4x − 9 + 18x−25
x2 −3
What are the four different ways that a function can be represented?
Verbal (sentences, function game), Algebraic, Numerical {(3,6)(2,9)etc}, Graph
Add
f (x) = (3x4 − 2x2 + 8x − 1)andg(x) = (5x3 + 4x2 − 9x + 12)
3x4 + 5x3 + 2x2 − x + 11
Subtract
f (x) = (4x3 − 8x2 + 5x − 13)andg(x) = (2x3 + 4x2 − 3x + 2)
2x3 − 12x2 + 8x − 15∗
M ultiply
f (x) = (4x3 − 5x2 + 2x − 3)andg(x) = (x − 9)
4x4 − 41x3 + 47x2 − 21x + 27
Divide
f (x) ÷ g(x)
Evaluate
g(x) = 12 − 4x
f (x) = 3x3 + 8x2 + 6x + 2andg(x) = (x2 + 2
g(y − 3)
g(7)
3x + 8 +
−16
−14
x2 +2
− 4y + 24
3x + 8
2
x +2
3
2
3x + 8x + 6x + 2
− 3x3
− 6x
8x2
− 8x2
+2
− 16
− 14
*3. When is f(x)+g(x) equal to zero?
Function can be written in words ”multiply by 2 and add 5”, or as an equation 2x+5
Write as an equation.
9. Triple the number.
10. Subtract 6
11. Square the number
1. f(x) = 3x
2. g(x) = x-6
3. h(x) = x2
Function Game. Write out each function as an equation then write the answer to each step. Don’t do these in your head
Start with x = 4
1. f(x) = 3x
2. f (x) = x + 1
3.
4.
5.
6.
7.
f (x) = x + 2
f (x) = x − 2
f (x) = x2
f (x) = x + 3
f (x) = −6x
8.
f (x) = x2
3. h(x) = x2
x
h(x) =
2
g(x) = x(x − 2)
h(x) = x + 6
g(x) = 3x
h(x) = x3
g(x) = x − 4
h(x) = 5x
g(x) = 4x
h(x) = x − 12
x
g(x) = x + 20
h(x) =
4
g(x) = 2x
h(x) = x − 1
2. g(x) = x-6
√
g(x) = x
1. Triple the number, then subtract six from the answer, then square the answer.
2. Add one. Then take the square root of the result. Then, divide that result into two.
3. Add two to the original number. Then, multiply by the original number. Then add 6 from the answer.
4. Subtract two, then triple the answer, the cube the answer.
5. Square, then subtract four from the answer, then multiply the answer by five
6. Add three. Then, multiply the answer by four. Then, subtract twelve from the answer. Then, divide by the original number.
7. Multiply by negative six. Add twenty to the answer. Divide the answer by four.
8. Square the number. Multiply the answer by 2. Subtract 1 from the answer.
Composite Functions
Take the answer from one function and plug it in to the next function.
1.
x = 4 → 12 → 6 → 36
2.
x=4→5→
3.
x = 4 → 6 → 24 → 30
4.
x = 4 → 2 → 6 → 216
5.
x = 4 → 16 → 12 → 60
6.
x = 4 → 7 → 28 → 140
7.
x = 4 → −24 → −4 → −1
8.
x = 4 → 16 → 32 → 31
√
√
5→
5
2
Function notation is 1. f(x) = 3x
2. g(x) = x-6
3. h(x) = x2
Composite Function Notation g(f(x)). Instead of plugging in x you plug in the answer for f(x).
EX 1:
g(x)=3x+3, f(x)=x2 + 2x + 1, Find g(f(-2))
Start on the inside
f(-2) = 1
Then plug in the answer to g(x)
g(1) = 6
Find g(f(5))
f(5) = 36
EX 2:
Find h(k(3)
k(3) = 7
Find k(h(-6))
h(3) = 5
g(36) = 111
h(x)=x2 − 4, k(x)=x2 − 2
h(7) = 45
k(5) =23
Find Practice Problems
f (x) = x2 + 4
EX 3
f (g(x))
h(x) =
√
x+5
show f o g(x) notation
f (g(x)) = 9x2 − 6x + 5
EX 4
Ex 5
g(x) = 3x − 1
Findf (h(x))
f (x) =
f (g(x)) =
3
x
3
2x−10
h(f (x)) =
p
x2 + 9
f (h(x)) = x + 9
g(x) = 2x − 10
6
g(f (x)) = − 10
x
Whiteboards on Composite Functions (DOMAIN and NON Numbers ON TUESDAY)
1.f (x) = 2x + 5
2.f (x) =
3.f (x) =
4.f (x) =
√
x+6
7
x−2
√
x−2
5.f (x) = 4x − 8
6.f (x) =
6
x
g(x) = x2 − 4
f (g(x)) = 2x2 − 3
g(x) = 3x − 9
f (g(x)) =
g(x) =
√
8−x
g(x) = 5x + 1
g(x) =
√
x+7
g(x) = x2 − 9
f (g(x)) =
f (g(x)) =
√
√ 7
8−x−2
√
f (g(x)) =
6
x2 −9
2. g(f(10)) = g(4) = 3
7
6
4. g(f(x)) = g(6) = 11
√
√
5. f(g(-4)) = f( 3) = 4 3 − 8
6. f(g(x) same as above
Exit Pass 1. f (x) = x2 − 6 and g(x) = 2x2 + 13x + 24 Find g(x) - f(x)
2. When is g(x) - f(x) equal to zero?
3. If f (x) =
√
x + 4 and g(x) = 5x + 1 find f(g(4))
5x − 1
√
f (g(x)) = 4 x + 7 − 8
1. f(g(8)) = f(60) = 125
3. f(g(-56) = f(8) =
3x − 3
g(f (x)) = 4x2 + 20x + 21
√
g(f (x)) = 3 x + 6 − 9
q
7
g(f (x)) = 8 − x−2
Domain : x 6= 2, x 6=
1
5
Domain : x ≥ −7
Domain : x 6= 0, x 6= ±3
1.f (x) = x2 − 3
1
x
Domain : x = ARN
g(x) = 2x − 1
h(x) =
f (g(x)) = 4x2 − 4x − 4
h(g(x)) =
f (h(x)) =
2.f (x) =
√
1
2x−1
1
x2 −
Domain : x 6=
1
2
1
2
Range : y 6= 0
Domain : x 6= 0
3
Range : y ≥
Rangey 6= −3
g(x) = −3x
h(x) = |x| − 12
√
Domain : x ≤ 0
Rangey ≥ 0
f (g(x)) = −6x
√
h(f (x)) − | 2x| − 12
Domain : x ≥ 0
Range : y ≥ −12
2x
g(h(x)) = −3|x| + 36
3.f (x) = x − 1
Domain : x = ARN
g(x) = x2 + 2x − 8
g(f (x)) = x2 − 9
f (h(x)) =
4.f (x) = −x + 1
2
x+1
−1=
g(x) =
2
x+1
Domain : x = ARN
1−x
x+1
Range : y ≥ −9
Domain : x 6= −1
p
x2 − 9
h(g(x)) = 3x2 − 27
p
g(f (x)) = x2 − 2x − 8
Deconstructing Functions
SKIP TO EX 5
h(x) =
Range : y ≤ 36
Range : y 6= −1
h(x) = 3x2
Domain : −3 ≤ x ≤ 3vRange : y ≥ 0
Domain : x ≤ −2 ∩ x ≥ 4
Range : y ≥ 0
EX 1:
EX 2:
EX 3:
f (g(x)) = (x + 1)2
√
f (g(x)) = x + 1
f (x) = x2
g(x) =?
f (x) = x − 1
2
f (g(x)) = (2x − 4) + 1
g(x) =?
g(x) = 2x − 4
f (x) =?
Show how to solve for g(x) when when given f(g(x))
EX 4:
f (x) = x − 5
f (g(x)) = x
(g(x)) − 5 = x
EX 5:
EX 5b:
g(x) =?
g(x) = x + 5
f (g(x)) = −x2 + 6
1
f (g(x)) =
4x − 2
g(x) = 4 − x2
f (x) = x + 2
f (x) =?
g(x) =?
several options for f(x) and g(x)
EX 5c:
EX 5d:
EX 5e:
EX 6:
EX 7:
1
f (x) =
x
p
f (g(x)) = 9x2 + 3
4
f (g(x)) =
10 − x2
p
f (g(x)) = x2 − 8
f (g(x)) = 2x − 3
1
f (g(x)) =
(x + 2)2
g(x) = 4x − 2
f (x) =?
f (x) =
g(x) =?
f (x) =?
g(x) =?
f (x) = x2
g(x) = 4x
g(x) =?
f (x) =?
f (x) = 2x + 3
1
f (x) =
x−2
1
x−2
g(x) = x − 3
g(x) = x2
g(x) =
1
x−2
1
g(x) = (x − 2)2
xp
f (x) = x2 + 1
g(x) = 2x − 5
f (x) =
EX 8:
9.
p
f (g(x)) =
4x2 − 20x + 26
f (x) = 4x − 5?
f (g(x)) = x
(g(x)) − 5 = x
10.
11.
12.
13.
14.
15.
16.
g(x) = x + 5
2
f (g(x)) = −x + 6
f (x) = 5x + 24 − x2
1
f (g(x)) =
f (x) = 5b??
4x − 2
several options for f(x) and g(x)
1
f (x) =
g(x) = 4x − 2
x
p
f (g(x)) = 9x2 + 3
f (x) = 5c??
4
f (g(x)) =
f (x) = 5d??
10 − x2
p
f (g(x)) = x2 − 8
f (x) = 5e??
f (g(x)) = 2x − 3
1
f (g(x)) =
(x + 2)2
f (g(x)) =
p
1
x−2
f (x) = 62x + 3x − 3
1
f (x) = 7
x2
x−2
f (x) = x2
17.
f (x) =
4x2 − 20x + 26
g(x) =
1
x−2
1
f (x) =
g(x) = (x − 2)2
x
p
f (x) = 8 x2 + 12x − 5
g(x) = 4x