1. Functions
Inverses

2. Starter: sketch each function
Functions Inverses
KUS objectives
BAT Find the inverse of functions
BAT understand the inverse of a function graphically
Starter: sketch each function
𝑦= 1 𝑥−2
𝑦= 1 𝑥−2 𝑥−4
𝑦= 𝑥−4

3. 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

4. 𝑓 −1 𝑥 =2(𝑥+8) 𝑔 −1 𝑥 = 𝑥−7 𝑓 −1 0 =16 𝑔 −1 7 =0 𝑓 −1 4 =24
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 =2(𝑥+8)
𝑔 −1 𝑥 = 𝑥−7
𝑓 −1 0 =16
𝑔 −1 7 =0
𝑓 −1 4 =24
𝑔 −1 15 =2 2
𝑔 −1 𝑥 3 = 𝑥 3 −7 = 𝑥−21 3
𝑓 −1 2𝑥 =2(2𝑥+8)=4(𝑥+4)

5. 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− 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

6. 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− 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

7. ⇒ 2𝑥𝑦−3𝑦=5𝑥+2 ⇒ 𝑥 2𝑦−5 =3𝑦+2 ⇒ 𝑦= 5𝑥+2 2𝑥−3 ⇒ 𝑦 2𝑥−3 =5𝑥+2
WB2c Find the inverse of each of these functions
a) 𝑓 𝑥 = 𝑥 2 3 −5 x∈𝑅 b) 𝑔 𝑥 = 6 3𝑥− 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=5𝑥+2
⇒ 𝑦= 5𝑥+2 2𝑥−3
⇒ 𝑦 2𝑥−3 =5𝑥+2
Step 3 write in function notation, write the domain if you know it
ℎ −1 𝑥 = 5𝑥+2 2𝑥−3 , 𝑥≠ 3 2

8. ⇒ 2𝑦+1=𝑎𝑟𝑐𝑠𝑖𝑛 3(𝑥+5) ⇒3(𝑥+5)= sin (2𝑦+1) ⇒ 𝑦= 𝑎𝑟𝑐𝑠𝑖𝑛 3𝑥+15 −1 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− 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 ,
−1≤(3𝑥+15)≤1
−14 3 ≤𝑥≤− 16 3

9. 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− 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= ln 𝑥
Step 3 write in function notation, write the domain if you know it
𝑔 −1 𝑥 = ln 𝑥 , 𝑥>0

10. 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− 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= − 𝑒 𝑥
Step 3 write in function notation, write the domain if you know it
𝑔 −1 𝑥 = − 𝑒 𝑥 , 𝑥∈𝑅

11. 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

12. 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 and 𝑔 −1 𝑥 = 𝑥
Use graph software
What do you notice?

13. 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

14. a) ⇒ ln 𝑦 𝑥 = 4𝑡 b) ⇒ 𝑒 2𝑥 = 𝑦 5 ⇒ 2𝑥= ln 𝑦 5 ⇒ y x = 𝑒 4𝑡
Think
Pair
Share
WB 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
⇒ 𝑥= ln 𝑦 5
⇒ y x = 𝑒 4𝑡
⇒ 𝑥= ln 𝑦 5
⇒ 𝑦= 𝑥 𝑒 4𝑡
⇒ 𝑥=𝑦 𝑒 −4𝑡

15. 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

16. 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

17. 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 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 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

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

19. 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 –

20. END