Functions composite

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

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 = 1.034 𝑥, 𝑥 > 0 A particular gas company’s charge in £ for t therms is given by the function g where 𝑔 𝑡 = 15 + 0.4 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) = 0.4136x + 15

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 …

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 2 a) find 𝑔𝑓 𝑥 and 𝑓𝑔 𝑥 b) find 𝑔𝑓 4 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

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

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 1 3 = 9 1 3 −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 = 1/3 9 = 1 27 𝑔𝑔 𝑥 = 𝑥 9

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 1 2 = 1 1 2 2 +2 = 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

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 = 1 6 2 − 2 6 +2= 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 −1 10 =10 𝑓 𝑓 −1 𝑥 =𝒙 of course!

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 5 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

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

Composite Domain and range

Sketch the graph of 𝑓𝑔 𝑥 and state its domain and range WB 6 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

Sketch the graph of 𝑔𝑓 𝑥 and state its domain and range WB 7 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

𝑔𝑓 𝑥 = 2𝑒 𝑓(𝑥) 𝑔𝑓 𝑥 = 2𝑒 ln 3𝑥−2 𝑔𝑓 𝑥 =2 3𝑥−2 =6𝑥 −4 WB 8 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

𝑓𝑔 𝑥 = 4𝑒 3 ln 𝑥 𝑓𝑔 𝑥 = 4𝑒 ln 𝑥 3 𝑓𝑔 𝑥 =4 𝑥 3 𝑥 ∈𝑅 WB 9 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 3 𝑥 ∈𝑅 Range 𝑓𝑔(𝑥)∈ℛ

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 1+ 5

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= 1 2 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 2 3 2

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 2 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− −6 2 2 =4− −32 2 =−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

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−10 x≥10 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 =3 d) find the inverse function 𝑓 −1 𝑥 a) Range 𝑓 𝑥 ≥0 b) 𝑓𝑔 𝑥 =𝑓 1 𝑥 = 1 𝑥 −10 c) 1 𝑥 −10 =3 ⇒ 1 𝑥 −10=9 ⇒ 𝑥= 1 19 𝑑) Let 𝑥= 𝑦−10 ⇒ 𝑓 −1 𝑥 = 𝑥 2 +10

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 –

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