INVERSE FUNCTION
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The document discusses functions and their inverses through examples. It begins with definitions of different types of functions and examples showing interchange of domain and range. It then works through examples of finding the inverse of different functions by interchanging domain and range. Homework is assigned involving finding the inverses of additional functions and evaluating the inverse at a point.
INVERSE FUNCTION
- 2. Review: Function • https://www.geogebra.org/m/mzxvxe44 INPUT Num. OUTPUT Num. (INPUT, OUTPUT) (3, 9) Domain Range 6/19/2020 JANAK SINGH SAUD 2 Click here
- 3. https://www.geogebra.org/m/vP9qTDF4 6/19/2020 JANAK SINGH SAUD 3 Click here
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- 5. Many to one-onto function 6/19/2020 JANAK SINGH SAUD 5 1 2 3 A B 5 After interchanging domain and range B 5 1 2 3 A It is not a function Many to one-into function 1 2 3 A B 3 4 5 After interchanging domain and range B 3 4 5 1 2 3 A It is not a function
- 6. One to one- into function 6/19/2020 JANAK SINGH SAUD 6 1 2 3 A B 1 4 9 16 After interchanging domain and range B 1 2 9 16 1 2 3 A It is not a function One to one- onto function 1 2 3 A B 1 4 9 After interchanging domain and range B 1 4 9 1 2 3 A It is function
- 7. 6/19/2020 JANAK SINGH SAUD 7 One to one- onto function 1 2 3 A B 1 4 9 B 1 4 9 1 2 3 Af f-1 Domain Range Domain Range Range of f = Domain of f-1 1 4 9 1 4 9 Domain of f = Range of f -1 1 2 3 1 2 3
- 8. 6/19/2020 JANAK SINGH SAUD 8 Range Domain Function (f) Inverse (f -1 ) Domain Range https://tasyaluftwascher.blogspot.com/2017/01/newest-for-images- drawing-cute-water.htmlSource
- 9. Conclusion : 6/19/2020 JANAK SINGH SAUD 9 A B B A f f-1 1 4 9 1 4 9 1 2 3 1 2 3 f(x) =y f(x) =yxx
- 10. Find the inverse of the function f(x) = x – 5 and show in arrow diagram also. • Here, y = f(x) = x – 5 • y = x – 5 • Interchanging domain (x) and range (y), we have • x = y - 5 • x + 5 = y • y = x + 5 ∴ f-1(x) = x + 5 6/19/2020 JANAK SINGH SAUD 10 EXAMPLE 1 1 2 3 A B - 4 -3 -2 B -4 -3 -2 1 2 3 A f f-1 • f(1) = 1 – 5 = - 4 • f(2) = 2 – 5 = - 3 • f(3) = 3 – 5 = - 2 • f – 1(- 4) = - 4 + 5 = 1 • f – 1( - 3) = - 3 + 5 = 2 • f – 1(- 2) = - 2 + 5 = 3 https://www.geogebra.org/m/mzxvxe44
- 11. Find the inverse of the function f(x) = 𝑥, 3𝑥−1 𝑥+1 : 𝑥 ∈ 𝑅 • Here, f(x) = y = 𝟑𝒙−𝟏 𝒙+𝟏 • y = 𝟑 𝒙 −𝟏 𝒙 +𝟏 • Interchanging domain (x) And range (y), we have • x = 𝟑 𝒚 −𝟏 𝒚 +𝟏 • 𝒙 𝟏 = 𝟑𝒚−𝟏 𝒚+𝟏 6/19/2020 JANAK SINGH SAUD 11 EXAMPLE 2 • 3y – 1 = xy + x • 3y – xy = x + 1 • (3 – x) y = x + 1 • Y = 𝒙+𝟏 𝟑 −𝒙 • ∴ f -1(x) = 𝒙+𝟏 𝟑 −𝒙
- 12. If f(x+ 3) = 5x + 4, then find f -1(x) • Here, f(x +3 ) = 5x+ 4 • Let x + 3 = k. Then • x = k – 3 • ∴ f(k) = 5(k – 3) + 4 • f(k) = 5k – 15 + 4 • f(k) = 5k – 11 • ∴ f(x) = 5x - 11 6/19/2020 JANAK SINGH SAUD 12 EXAMPLE 3 • Again, y = f(x) = 5x – 11 • y = 5 x – 11 • Interchanging domain (x) and range (y), we get • x = 5 y – 11 • x + 11 = 5y • 𝒙+𝟏𝟏 𝟓 = y • y = 𝒙+𝟏𝟏 𝟓 • ∴ f -1(x) = 𝒙+𝟏𝟏 𝟓 Replace k = x
- 13. Home work • Find the inverse of the following functions and show in the arrow diagrams 1. f(x) = 5 – 7x 2. g(x) = 3𝑥+1 4 3. h(x) = {(1, 1), (2, 4), (3, 9), (4, 16)}. 4. f(x) = {(1, 3), (2, 3),(3, 5),(4, 6),(5, 7}. if exists. 5. if f(x) = 𝑥+1 𝑥 −1 find f-1(2) 6/19/2020 JANAK SINGH SAUD 13