Functions Inverses

Functions Inverses Starter: KUS objectives BAT Find the inverse of functions BAT understand the inverse of a function graphically Starter:

An Inverse function is an ‘opposite’ function Notes An Inverse function is an ‘opposite’ function e.g. The inverse of f (x) is written as: f-1(x) and

𝑓 −1 𝑥 = 𝑥+8 2 𝑔 −1 𝑥 = 𝑥−7 𝑓 −1 0 =4 𝑔 −1 7 =0 𝑓 −1 4 =6 𝑔 −1 15 =2 2 WB1 f and g are functions defined by 𝑓 𝑥 = 1 2 𝑥−8 and 𝑔 𝑥 = 𝑥 2 +7 Find the inverse functions 𝑓 −1 (𝑥) and 𝑔 −1 (𝑥) and find a) 𝑓 −1 (0) 𝑓 −1 (4) 𝑓 −1 (2𝑥) and b) 𝑔 −1 (7) 𝑔 −1 (15) 𝑔 −1 𝑥 3 × 1 2 −8 +8 ÷2 𝑥 2 +7 −7 𝑥 𝑓 −1 𝑥 = 𝑥+8 2 𝑔 −1 𝑥 = 𝑥−7 𝑓 −1 0 =4 𝑔 −1 7 =0 𝑓 −1 4 =6 𝑔 −1 15 =2 2 𝑔 −1 𝑥 3 = 𝑥 3 −7 = 𝑥−21 3 𝑓 −1 2𝑥 = 2𝑥+8 2 =𝑥+4

𝑥= 𝑦 2 3 −5 ⇒ 𝑦 2 =3(𝑥+5) ⇒ 𝑦= 3(𝑥−5) 𝑓 −1 𝑥 = 3(𝑥−5) , 𝑥>5 WB2a SWAPPING METHOD Find the inverse of each of these functions 𝑓 𝑥 = 𝑥 2 3 −5 x∈𝑅 b) 𝑔 𝑥 = 6 3𝑥−2 c) ℎ 𝑥 = 3𝑥+2 2𝑥−5 𝑥= 𝑦 2 3 −5 Step 1 swap x and y around Step 2 rearrange until you get y = … ⇒ 𝑦 2 =3(𝑥+5) ⇒ 𝑦= 3(𝑥−5) Step 3 write in function notation, write the domain if you know it 𝑓 −1 𝑥 = 3(𝑥−5) , 𝑥>5

𝑥 3𝑦−2 =6 ⇒ 3𝑥𝑦−2𝑥=6 ⇒ 𝑦= 2𝑥+6 3𝑥 ⇒ 3𝑦𝑥=2x+6 𝑔 −1 𝑥 = 2𝑥+6 3𝑥 , 𝑥≠0 WB2b Find the inverse of each of these functions a) 𝑓 𝑥 = 𝑥 2 3 −5 x∈𝑅 b) 𝑔 𝑥 = 6 3𝑥−2 c) ℎ 𝑥 = 3𝑥+2 2𝑥−5 𝑥= 6 3𝑦−2 Step 1 swap x and y around Step 2 rearrange until you get y = … 𝑥 3𝑦−2 =6 ⇒ 3𝑥𝑦−2𝑥=6 ⇒ 3𝑦𝑥=2x+6 ⇒ 𝑦= 2𝑥+6 3𝑥 Step 3 write in function notation, write the domain if you know it 𝑔 −1 𝑥 = 2𝑥+6 3𝑥 , 𝑥≠0

⇒ 2𝑥𝑦−3𝑦=2+5𝑥 ⇒ 𝑥 2𝑦−5 =3𝑦+2 ⇒ 𝑦= 5𝑥+2 2𝑥−3 ⇒ 𝑦(2𝑥−3)=2+5𝑥 WB2c Find the inverse of each of these functions a) 𝑓 𝑥 = 𝑥 2 3 −5 x∈𝑅 b) 𝑔 𝑥 = 6 3𝑥−2 c) ℎ 𝑥 = 3𝑥+2 2𝑥−5 𝑥= 3𝑦+2 2𝑦−5 Step 1 swap x and y around Step 2 rearrange until you get y = … ⇒ 𝑥 2𝑦−5 =3𝑦+2 ⇒ 2𝑥𝑦−3𝑦=2+5𝑥 ⇒ 𝑦= 5𝑥+2 2𝑥−3 ⇒ 𝑦(2𝑥−3)=2+5𝑥 Step 3 write in function notation, write the domain if you know it ℎ −1 𝑥 = 5𝑥+2 2𝑥−3 , 𝑥≠0

𝑓 −1 𝑥 = 𝑎𝑟𝑐𝑠𝑖𝑛 3𝑥+15 −1 2 , 4 3 <𝑥<2 WB 3a Find the inverse of each of these functions a) 𝑓 𝑥 = sin (2𝑥+1) 3 −5 x∈𝑅 b) 𝑔 𝑥 = 𝑒 3𝑥−2 c) ℎ 𝑥 = ln 4−6𝑥 𝑥= sin (2𝑦+1) 3 −5 Step 1 swap x and y around Step 2 rearrange until you get y = … ⇒3(𝑥+5)= sin (2𝑦+1) ⇒ 2𝑦+1=𝑎𝑟𝑐𝑠𝑖𝑛 3(𝑥+5) ⇒ 𝑦= 𝑎𝑟𝑐𝑠𝑖𝑛 3𝑥+15 −1 2 Step 3 write in function notation, write the domain if you know it 𝑓 −1 𝑥 = 𝑎𝑟𝑐𝑠𝑖𝑛 3𝑥+15 −1 2 , 4 3 <𝑥<2

𝑥= 𝑒 3𝑦−2 ⇒ ln 𝑥 =3𝑦−2 ⇒ 𝑦= 1 3 2+ ln 𝑥 𝑔 −1 𝑥 = 1 3 2+ ln 𝑥 , 𝑥>0 WB 3b Find the inverse of each of these functions a) 𝑓 𝑥 = sin (2𝑥+1) 3 −5 x∈𝑅 b) 𝑔 𝑥 = 𝑒 3𝑥−2 c) ℎ 𝑥 = ln 4−6𝑥 𝑥= 𝑒 3𝑦−2 Step 1 swap x and y around Step 2 rearrange until you get y = … ⇒ ln 𝑥 =3𝑦−2 ⇒ 𝑦= 1 3 2+ ln 𝑥 Step 3 write in function notation, write the domain if you know it 𝑔 −1 𝑥 = 1 3 2+ ln 𝑥 , 𝑥>0

𝑥= ln 4−6𝑦 ⇒ 𝑒 𝑥 =4 −6𝑦 ⇒ 𝑦= 1 6 4− 𝑒 𝑥 𝑔 −1 𝑥 = 1 6 4− 𝑒 𝑥 , 𝑥∈𝑅 WB 3c Find the inverse of each of these functions a) 𝑓 𝑥 = sin (2𝑥+1) 3 −5 x∈𝑅 b) 𝑔 𝑥 = 𝑒 3𝑥−2 c) ℎ 𝑥 = ln 4−6𝑥 𝑥= ln 4−6𝑦 Step 1 swap x and y around Step 2 rearrange until you get y = … ⇒ 𝑒 𝑥 =4 −6𝑦 ⇒ 𝑦= 1 6 4− 𝑒 𝑥 Step 3 write in function notation, write the domain if you know it 𝑔 −1 𝑥 = 1 6 4− 𝑒 𝑥 , 𝑥∈𝑅

𝑎) 𝑓 𝑥 = (𝑥−3) 2 −1 b) 𝑔 𝑥 =3 (𝑥−2) 2 +6 Let 𝑥= (𝑦−3) 2 −1 WB 4 Rearrange each function to complete square form then find its inverse a) 𝑓 𝑥 = 𝑥 2 +6𝑥+8 x∈𝑅 b) 𝑔 𝑥 = 3𝑥 2 −12𝑥+18 𝑎) 𝑓 𝑥 = (𝑥−3) 2 −1 b) 𝑔 𝑥 =3 (𝑥−2) 2 +6 Let 𝑥= (𝑦−3) 2 −1 Let 𝑥=3 (𝑦−2) 2 +6 (𝑦−3) 2 =𝑥+1 (𝑦−2) 2 = 𝑥−6 3 𝑦−3= 𝑥+1 𝑦−2= 𝑥−6 3 𝑦=3+ 𝑥+1 𝑔 −1 𝑥 =2+ 𝑥−6 3 , 𝑥≥6 𝑓 −1 𝑥 =3+ 𝑥+1 , 𝑥≥−1

Think Pair 𝑓 𝑥 =2𝑥+1 and 𝑓 −1 𝑥 = 𝑥−1 2 share Use graph software WB 5 Sketch each pair of functions on the same set of axes 𝑓 𝑥 =2𝑥+1 and 𝑓 −1 𝑥 = 𝑥−1 2 𝑔 𝑥 = 𝑥 2 and 𝑔 −1 𝑥 = 𝑥 Use graph software What do you notice?

𝑓 𝑥 Domain 𝒙<𝟖 Range 𝒚 𝝐 𝑹 Notes 𝑓 𝑥 Domain 𝒙<𝟖 Range 𝒚 𝝐 𝑹 Domain of f is range of its inverse … range of f is domain of its inverse 𝑓 −1 𝑥 Domain 𝒙 𝝐 𝑹 Range 𝒚<𝟖

a) ⇒ ln 𝑦 𝑥 = 4𝑡 b) ⇒ 𝑒 2𝑥 = 𝑦 5 ⇒ 2𝑥= ln 𝑦 5 ⇒ y x = 𝑒 4𝑡 Think Pair Share WB6 a) Make x the subject of: 𝑙𝑛𝑦 −𝑙𝑛𝑥=4𝑡 b) Make x the subject of: 𝑦 = 5𝑒 2𝑥 a) ⇒ ln 𝑦 𝑥 = 4𝑡 b) ⇒ 𝑒 2𝑥 = 𝑦 5 ⇒ 2𝑥= ln 𝑦 5 ⇒ y x = 𝑒 4𝑡 ⇒ 𝑥= 1 2 ln 𝑦 5 ⇒ 𝑦= 𝑥 𝑒 4𝑡 ⇒ 𝑥=𝑦 𝑒 −4𝑡

WB7 The function f is defined by 𝑓:𝑥→ ln 3𝑥−2 , 𝑥∈ℛ, 𝑥≥ 2 3   Find 𝑓 −1 (𝑥) State the domain of 𝑓 −1 (𝑥) 𝑦= 2 3 𝑓 𝑥 = ln 3𝑥−2 Let 𝑥= ln 3𝑦−2 𝑒 𝑥 = 3𝑦 −2 𝑦 = 𝑒 𝑥 +2 3 𝑓 −1 (𝑥) = 𝑒 𝑥 +2 3 Domain 𝑥∈ℛ

WB8 The function f is defined by 𝑓:𝑥→ 2𝑒 𝑥+1   Find 𝑓 −1 (𝑥) State the domain of 𝑓 −1 (𝑥) 𝑓 𝑥 = 2𝑒 𝑥+1 Let 𝑥= 2𝑒 𝑦+1 𝑒 𝑦+1 = 𝑥 2 Domain 𝑥∈ℛ, 𝑥>0 𝑦+1= ln 𝑥 2 𝑓 −1 𝑥 = ln 𝑥 2 −1 Domain 𝑥∈ℛ, 𝑥>0

a) = 4 𝑥+7 𝑥+7 + 3 𝑥+7 = 4𝑥+31 𝑥+7 b) Let x= 4𝑦+31 𝑦+7 ⇒ x y+7 =4𝑦+31 WB9 exam Q 𝑓 𝑥 =4+ 3 𝑥+7 x∈𝑅, 𝑥≠7 a) Express 4+ 3 𝑥+7 s a single fraction Find an expression for 𝑓 −1 𝑥 write the domain of 𝑓 −1 𝑥 a) = 4 𝑥+7 𝑥+7 + 3 𝑥+7 = 4𝑥+31 𝑥+7 b) Let x= 4𝑦+31 𝑦+7 ⇒ x y+7 =4𝑦+31 ⇒ 𝑥𝑦−4𝑦=31−7𝑥 ⇒ 𝑦= 31−7𝑥 𝑥−4 c) Domain 𝑥∈𝑅 𝑥≠4

a) Let x=1+ 3 𝑦−3 ⇒ (𝑥−1) 2 = 3 𝑦+3 ⇒ 𝑦+3= 3 (𝑥−1) 2 ⇒ 𝑦= 3 (𝑥−1) 2 −3 WB10 exam Q 𝑓 𝑥 =1+ 3 𝑥−3 x∈𝑅, 𝑥≠3 a) Find an expression for 𝑓 −1 𝑥 write the domain of 𝑓 −1 𝑥 a) Let x=1+ 3 𝑦−3 ⇒ (𝑥−1) 2 = 3 𝑦+3 ⇒ 𝑦+3= 3 (𝑥−1) 2 ⇒ 𝑦= 3 (𝑥−1) 2 −3 b) Domain 𝑥∈𝑅 𝑥≠1

BAT understand the inverse of a function graphically KUS objectives BAT Find the inverse of functions BAT understand the inverse of a function graphically self-assess One thing learned is – One thing to improve is –

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