1. Functions
composite

2. Starter: find the inverse functions of
Composite functions
KUS objectives
BAT Find composite functions
BAT Find the domain and range of composite functions
Starter: find the inverse functions of
𝑓 𝑥 = 1 𝑥−2
𝑔 𝑥 =2 (𝑥−7) 2
ℎ 𝑥 = ( sin 𝑥 −1) 2 +3

3. Find the cost of using 100 cubic feet
WB1
A gas meter indicates the amount of gas in cubic feet used by a consumer. The number of therms of heat from x cubic feet of gas is given by the function f where
𝑓 𝑥 = 𝑥, 𝑥 > 0
A particular gas company’s charge in £ for t therms is given by the function g where
𝑔 𝑡 = 𝑡
Find the cost of using 100 cubic feet
Find a single rule for working out the cost given the number of cubic feet of gas used
£56.36
gf(x) = x + 15

4. A composite function is a ‘function of a function’ e.g.
Notes
A composite function is a ‘function of a function’
e.g.
𝒇 𝒙 =𝟐𝒙+𝟏
𝒈 𝒙 = 𝒙 𝟐 𝒙 3 5 7 9 …
𝒇(𝒙) 9 25 49 81 …
𝒈𝒇(𝒙) 1 2 3 4 …

5. 𝒇 𝒙 =𝟐𝒙+𝟏 𝒈 𝒙 = 𝒙 𝟐 1 2 3 4 … 𝒙 5 7 9 𝒇(𝒙) 25 49 81 𝒈𝒇(𝒙)
WB2 𝑓 𝑥 =2𝑥+1 and 𝑔 𝑥 = 𝑥 a) find 𝑔𝑓 𝑥 and 𝑓𝑔 𝑥
b) find 𝑔𝑓 and 𝑓𝑔 4 1 2 3 4 … 𝒙 5 7 9
𝒇(𝒙) 25 49 81
𝒈𝒇(𝒙)
𝒇 𝒙 =𝟐𝒙+𝟏
𝒈 𝒙 = 𝒙 𝟐
𝒈𝒇 𝒙 = 𝒈 𝟐𝒙+𝟏 = 𝟐𝒙+𝟏 𝟐
𝒈𝒇 𝟒 =𝒈 𝟗 = 𝟖𝟏
and
𝒇𝒈 𝒙 =𝒇 𝒙 𝟐 =𝟐 𝒙 𝟐 +𝟏
𝒇𝒈 𝟒 =𝒇 𝟏𝟔 =𝟑𝟑

6. a) 𝑔𝑓 𝑥 𝑔𝑓 0 b) 𝑓𝑔 𝑥 𝑓𝑔 0 Find: 𝑓𝑔 𝑥 =3 𝒙 𝟑 −1 𝑓𝑔 0 =0−1=−1 𝑓𝑔 𝑥 =𝑥−1
WB3ab 𝒇 𝒙 =𝟑𝒙−𝟏 𝒈 𝒙 = 𝒙 𝟑
Find:
a) 𝑔𝑓 𝑥 𝑔𝑓 0
𝑔𝑓 𝑥 = (3𝑥−1) 3
𝑔𝑓 0 = (3(0)−1) 3 =− 1 3
b) 𝑓𝑔 𝑥 𝑓𝑔 0
𝑓𝑔 𝑥 =3 𝒙 𝟑 −1
𝑓𝑔 𝑥 =𝑥−1
𝑓𝑔 0 =0−1=−1

7. c) 𝑓𝑓 𝑥 𝑓𝑓 1 3 d) 𝑔𝑔 𝑥 𝑔𝑔 1 3 Find: 𝑓𝑓 𝑥 =3(3𝑥−1)−1 𝑓𝑓 𝑥 =9𝑥−4
WB3cd 𝒇 𝒙 =𝟑𝒙−𝟏 𝒈 𝒙 = 𝒙 𝟑
Find:
c) 𝑓𝑓 𝑥 𝑓𝑓 1 3
𝑓𝑓 = −4=−1
𝑓𝑓 𝑥 =3(3𝑥−1)−1
𝑓𝑓 𝑥 =9𝑥−4
d) 𝑔𝑔 𝑥 𝑔𝑔 1 3
𝑔𝑔 𝑥 = 𝒙/𝟑 𝟑
𝑔𝑔 = 1/3 9 = 1 27
𝑔𝑔 𝑥 = 𝑥 9

8. a) 𝑔𝑓 𝑥 𝑔𝑓 1 2 b) 𝑓𝑔 𝑥 𝑓𝑔 −3 Find: 𝑔𝑓 𝑥 = 𝟏 ( 𝒙 𝟐 +𝟏)+𝟏
WB4ab 𝒇 𝒙 = 𝒙 𝟐 +𝟏 𝒈 𝒙 = 𝟏 𝒙+𝟏
Find:
a) 𝑔𝑓 𝑥 𝑔𝑓 1 2
𝑔𝑓 𝑥 = 𝟏 ( 𝒙 𝟐 +𝟏)+𝟏
𝑔𝑓 = = 4 9
𝑔𝑓 𝑥 = 1 𝑥 2 +2
b) 𝑓𝑔 𝑥 𝑓𝑔 −3
𝑓𝑔 𝑥 = 𝟏 𝒙+𝟏 𝟐 +𝟏
𝑓𝑔 −3 = 𝟏 −𝟑+𝟏 𝟐 +𝟏= 𝟓 𝟒
𝑓𝑔 𝑥 = 𝟏 𝒙+𝟏 𝟐 +𝟏

9. c) 𝑓 𝑔 −1 𝑥 𝑓 𝑔 −1 6 d) 𝑓 𝑓 −1 𝑥 𝑓 𝑓 −1 10 Find:
WB4cd 𝒇 𝒙 = 𝒙 𝟐 +𝟏 𝒈 𝒙 = 𝟏 𝒙+𝟏
Find:
𝒈 −𝟏 𝒙 = 𝟏 𝒙 −𝟏
𝒇 −𝟏 𝒙 = 𝒙−𝟏
c) 𝑓 𝑔 −1 𝑥 𝑓 𝑔 −1 6
𝑓 𝑔 −1 𝑥 = 𝟏 𝒙 −𝟏 𝟐 +𝟏
𝑔𝑓 6 = − = 61 36
𝑓 𝑔 −1 𝑥 = 1 𝑥 2 − 2 𝑥 +2
d) 𝑓 𝑓 −1 𝑥 𝑓 𝑓 −1 10
𝑓 𝑓 −1 𝑥 = 𝒙−𝟏 𝟐 +𝟏
𝑓 𝑓 − =10
𝑓 𝑓 −1 𝑥 =𝒙 of course!

10. 1 2𝑥+1 𝑥 8− 1 𝑥 1 8 −𝑥 (8−𝑥) 2 𝒌𝒈 𝒙 𝒇𝒉 𝒙 𝒌𝒌 𝒙 2 𝑥 2 +1 4 𝑥 2 +4𝑥+1
WB 𝒇 𝒙 = 𝒙 𝟐 𝒈 𝒙 =𝟐𝒙+𝟏
𝒉 𝒙 =𝟖−𝒙 𝒌 𝒙 = 𝟏 𝒙
Find the composite functions that correspond to:
1 2𝑥+1
(8−𝑥) 2 𝑥
𝒌𝒈 𝒙
𝒇𝒉 𝒙
𝒌𝒌 𝒙
8− 1 𝑥
2 𝑥 2 +1
4 𝑥 2 +4𝑥+1
𝒉𝒌 𝒙
𝒈𝒈 𝒙
𝒈𝒇 𝒙
1 8 −𝑥
𝑥 4
7− 2𝑥 2
𝒇𝒇 𝒙
𝒌𝒉 𝒙
𝒉𝒈𝒇 𝒙

11. Challenge work in pairs
Choose four integers between 1 and 9 inclusive,
called a, b, c, d
Write the functions 𝑓 𝑥 = 𝑒 𝑎𝑥+𝑏 and 𝑔 𝑥 = ln (𝑐𝑥+𝑑)
Find a) fg(x) b) the inverse … of your neighbours functions

12. Composite
Domain and range

13. Sketch the graph of 𝑓𝑔 𝑥 and state its domain and range
WB 𝒇 𝒙 = 𝒙 𝟐 𝒈 𝒙 =𝒙+𝟑
Sketch the graph of 𝑓𝑔 𝑥 and state its domain and range
𝒇𝒈 𝒙 = (𝒙+𝟑) 𝟐
Domain 𝒙 𝝐 𝑹
Range 𝒚>𝟎

14. Sketch the graph of 𝑔𝑓 𝑥 and state its domain and range
WB 𝒇 𝒙 =𝒙−𝟐 𝒈 𝒙 = 𝟑𝒙
Sketch the graph of 𝑔𝑓 𝑥 and state its domain and range
𝒈𝒇 𝒙 = 𝟑(𝒙−𝟐)
𝒈𝒇 𝒙 = 𝟑𝒙−𝟔
Domain 𝒙>𝟐
Range 𝒚>𝟎

15. 𝑔𝑓 𝑥 = 2𝑒 𝑓(𝑥) 𝑔𝑓 𝑥 = 2𝑒 ln 3𝑥−2 𝑔𝑓 𝑥 =2 3𝑥−2 =6𝑥 −4
WB The function f is defined by 𝑓:𝑥→ ln 3𝑥−2 , 𝑥∈ℛ, 𝑥≥ 2 3
 The function g is defined by 𝑔:𝑥→ 2𝑒 𝑥 , 𝑥∈ℛ
a) Find 𝑔𝑓(𝑥), giving your answer in its simplest form
b) Find the range of 𝑔𝑓(𝑥)
𝑔𝑓 𝑥 = 2𝑒 𝑓(𝑥)
𝑔𝑓 𝑥 = 2𝑒 ln 3𝑥−2
𝑔𝑓 𝑥 =2 3𝑥−2 =6𝑥 −4
This is a linear graph
Range 𝑓𝑔(𝑥)∈ℛ

16. 𝑓𝑔 𝑥 = 4𝑒 3 ln 𝑥 𝑓𝑔 𝑥 = 4𝑒 ln 𝑥 3 𝑓𝑔 𝑥 =4 𝑥 3 𝑥 ∈𝑅
WB The function f has domain [−∞, ∞] and is defined by 𝑓 𝑥 =4 𝑒 𝑥
The function g has domain [1, ∞] and is defined by 𝑔 𝑥 =3 ln 𝑥
Explain why 𝑓𝑔 −3 does not exist
Find in its simplest form an expression for 𝑓𝑔 𝑥 stating its domain and range
𝑓𝑔 −3 =𝑓( 3 ln (−3) ) but ln (-3) is not possible
𝑓𝑔 𝑥 = 4𝑒 3 ln 𝑥
Rules of logarithms
𝑓𝑔 𝑥 = 4𝑒 ln 𝑥 3
𝑓𝑔 𝑥 =4 𝑥 𝑥 ∈𝑅
Range 𝑓𝑔(𝑥)∈ℛ

17. 𝒈 −𝟏 𝒙 = 𝒆 𝒙 −𝟏 a) 𝑔𝑓 𝑥 = 6 b) 𝑓 𝑔 −1 𝑥 =2 Solve: 𝑔𝑓 𝑥 =𝒍𝒏 𝒙 𝟐 −𝟑 +𝟏
WB10ab 𝒇 𝒙 = 𝒙 𝟐 −𝟑 𝒈 𝒙 =𝒍𝒏 (𝒙+𝟏)
Solve:
𝒈 −𝟏 𝒙 = 𝒆 𝒙 −𝟏
a) 𝑔𝑓 𝑥 = 6
𝑔𝑓 𝑥 =𝒍𝒏 𝒙 𝟐 −𝟑 +𝟏
So 𝑔𝑓 𝑥 = ln 𝑥 2 −4 =6
𝑥 2 −4 = 𝑒 6
𝑥= 𝑒 6 +4
b) 𝑓 𝑔 −1 𝑥 =2
𝑓 𝑔 −1 𝑥 = 𝒆 𝒙 −𝟏 𝟐 −𝟑
So 𝑓 𝑔 −1 𝑥 = 𝑒 𝑥 −1 2 −3=2
𝑒 𝑥 −1 2 =5
𝑒 𝑥 =1+ 5
𝑥= ln

18. 𝒇 −𝟏 𝒙 = 𝟑−𝒙 a) 𝑓𝑔 𝑥 =1 b) 𝑔𝑓 −1 𝑥 =2 Solve: 𝑓𝑔 𝑥 = 𝟑− 𝟑 𝒆 𝒙 𝟐
WB11ab 𝒇 𝒙 = 𝟑− 𝒙 𝟐 𝒈 𝒙 =𝟑 𝒆 𝒙
Solve:
𝒇 −𝟏 𝒙 = 𝟑−𝒙
a) 𝑓𝑔 𝑥 =1
𝑓𝑔 𝑥 = 𝟑− 𝟑 𝒆 𝒙 𝟐
So 𝑔𝑓 𝑥 =3−9 𝑒 2𝑥 =1
𝑒 2𝑥 = 2 9
𝑥= ln 2 9
b) 𝑔𝑓 −1 𝑥 =2
𝑔𝑓 −1 𝑥 =𝟑 𝒆 𝟑−𝒙
So 𝑓 𝑔 −1 𝑥 =𝟑 𝒆 𝟑−𝒙 =2
𝟑−𝒙 = ln 2 3
𝑥=3− ln

19. 𝑓 −1 𝑥 = 4−𝑥 , 𝑥<4 a) ff −6 =4− 4− −6 2 2 =4− −32 2 =−1020
WB12 exam Q 𝑓 𝑥 =4− 𝑥 x≤0
Evaluate 𝑓𝑓(−6) F
Find an expression for 𝑓 −1 𝑥
Write the domain and range of 𝑓 −1 𝑥
a) ff −6 =4− 4− − =4− − =−1020
b) Let x=4− 𝑦 2
⇒ 𝑦 2 =4−𝑥
⇒ 𝑦= 4−𝑥
𝑓 −1 𝑥 = 4−𝑥 , 𝑥<4
c) Domain 𝑥<4 range 𝑥≤0

20. a) Range 𝑓 𝑥 ≥0 b) 𝑓𝑔 𝑥 =𝑓 1 𝑥 = 1 𝑥 −10 c) 1 𝑥 −10 =3 ⇒ 1 𝑥 −10=9
WB13 exam Q the functions f and g are defined by
𝑓 𝑥 = 𝑥− x≥ g 𝑥 = 1 𝑥 x∈𝑅, 𝑥≠0
a) State the range of f(x) b) Find 𝑓𝑔 𝑥
c) Solve the equation 𝑓𝑔 𝑥 = d) find the inverse function 𝑓 −1 𝑥
a) Range 𝑓 𝑥 ≥0
b) 𝑓𝑔 𝑥 =𝑓 1 𝑥 = 1 𝑥 −10
c) 𝑥 −10 =3
⇒ 𝑥 −10=9
⇒ 𝑥= 1 19
𝑑) Let 𝑥= 𝑦−10
⇒ 𝑓 −1 𝑥 = 𝑥 2 +10

21. One thing to improve is –
KUS objectives
BAT Find composite functions
BAT Find the domain and range of composite functions
self-assess
One thing learned is –
One thing to improve is –

22. END