1. Functions
Review

2. Choose two numbers from 1 to 8 and two letters from A to H
Let the numbers be functions f(x) and g(x)
And the letters be corresponding transformations of these functions

3. 𝐴 𝑥 = 𝑥 2 +2 𝑓 𝑥+3 𝐴 𝑥 = 𝑥 2 +𝑥 𝑓 2𝑥 𝐴 𝑥 = −𝑥 2 2𝑓 𝑥 𝐴 𝑥 = 1 𝑥 𝑓 𝑥−2
FUNCTION
TRANSFORMATION
𝐴 𝑥 = 𝑥 2 +2
𝐴 𝑥 = 𝑥 2 +𝑥
𝐴 𝑥 = −𝑥 2
𝐴 𝑥 = 1 𝑥
𝐴 𝑥 = 2𝑥 2 −1
𝐴 𝑥 = 1 𝑥 +3
𝐴 𝑥 = 3−𝑥 2
𝐴 𝑥 = 1 𝑥+1
𝑓 𝑥+3
𝑓 2𝑥
2𝑓 𝑥
𝑓 𝑥−2
𝑓 𝑥 +2
𝑓 𝑥 −1
𝑓 1 2 𝑥
𝑓 𝑥+2

4. Sketch graphs of each function
state the domain and inverse and show key points
2) Find the inverse of each function
3) Sketch graphs of each inverse function
state the domain and inverse and show key points
4) Apply the transformations to each function algebraically
5) Sketch graphs of each transformed function
state the domain and inverse and show key points
6) Work out the composite functions fg(x) and gf(x) state the domain and inverse of composite functions

5. Sketch graphs of each function
Example
𝑓 𝑥 = 𝑥 2 +2
𝑔 𝑥 = 1 𝑥 +3
A: 𝑓 𝑥+3
F: 𝑔 𝑥 −1
Sketch graphs of each function
state the domain and inverse and show key points

6. 2) Find the inverse of each function
Example
𝑓 𝑥 = 𝑥 2 +2
𝑔 𝑥 = 1 𝑥 +3
2) Find the inverse of each function
x= 𝑦 2 +2
x= 1 𝑦 +3
𝑦 2 =𝑥−2
1 𝑦 =𝑥−3
𝑦=+ 𝑥−2
𝑦= 1 𝑥−3
𝑓 −1 (𝑥)= 𝑥−2
𝑔 −1 𝑥 = 1 𝑥−3

7. 3) Sketch graphs of each inverse function
Example
𝑓 𝑥 = 𝑥 2 +2
𝑔 𝑥 = 1 𝑥 +3
3) Sketch graphs of each inverse function
state the domain and inverse and show key points
𝑓 −1 (𝑥)= 𝑥−2
𝑔 −1 𝑥 = 1 𝑥−3

8. 4) Apply the transformations to each function algebraically
Example
𝑓 𝑥 = 𝑥 2 +2
𝑔 𝑥 = 1 𝑥 +3
A: 𝑓 𝑥+3
F: 𝑔 𝑥 −1
4) Apply the transformations to each function algebraically
𝑔 𝑥 −1 = 1 𝑥 +3 −1
𝑓 𝑥+3 = (𝑥+3) 2 +2
𝑓 𝑥+3 = 𝑥 2 +6𝑥+11
𝑔 𝑥 −1 = 1 𝑥 +2

9. 5) Sketch graphs of each transformed function
Example
𝑓 𝑥 = 𝑥 2 +2
𝑔 𝑥 = 1 𝑥 +3
5) Sketch graphs of each transformed function
state the domain and inverse and show key points
𝑓 𝑥+3 = 𝑥 2 +6𝑥+11
𝑔 𝑥 −1 = 1 𝑥 +2

10. Example
𝑓 𝑥 = 𝑥 2 +2
𝑔 𝑥 = 1 𝑥 +3
6) Work out the composite functions fg(x) and gf(x) state the domain and inverse of composite functions
𝑓 𝑔(𝑥) = 𝑓 1 𝑥 +3
𝑔 𝑓(𝑥) = 𝑓 𝑥 2 +2
= 1 𝑥
= 1 𝑥
DOMAIN: 𝑥∈ℛ
RANGE: graph it to see
= 1 𝑥 𝑥 +11
DOMAIN: 𝑥≠0, 𝑥∈ℛ
RANGE: graph it to see

11. END