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.

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6/19/2020 JANAK SINGH SAUD 1

2. Review: Function
• https://www.geogebra.org/m/mzxvxe44
INPUT Num.
OUTPUT Num.
(INPUT, OUTPUT)
(3, 9)
Domain
Range
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3. https://www.geogebra.org/m/vP9qTDF4
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4. 6/19/2020 JANAK SINGH SAUD 4

5. Many to one-onto function
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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
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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

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Range
Domain
Function (f)
Inverse (f -1 )
Domain
Range
https://tasyaluftwascher.blogspot.com/2017/01/newest-for-images-
drawing-cute-water.htmlSource

9. Conclusion
:
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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
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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
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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
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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)
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