Consider the following functions For each function determine

Consider the following functions. For each function, determine if it is 1-1, onto, and invertible. If a function is invertible, provide the inverse function. Justify each of your answers (modified from/.y Books exercise 4.3.1). Let A_5 = {1, 2, 3, 4, 5, 6, 7, 8}. Let f_5: P(A) rightarrow {0, 1, 2, 3, 4, 5, 6, 7, 8} be the function defined by f_5(X) = |X| (note that X A). Let f_6: {a, b}^3 rightarrow {a, b}^3 defined by f_6{x, y, z) = (z, y, x).

Solution

Domain = Set of possible values to be plugged into the function.

Co-domain = Set of possible values that may be the output of the function.

Range = Set of actual values that are the result of the function.

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(a)

X

1

2

3

4

5

6

7

8

F(X)

1

2

3

4

5

6

7

8

Co-domain = {0,1,2,3,4,5,6,7,8}

Since, |X| is always unique for X > 0. (For negative values, two values of X may have same value of |X| but there are no negative values in our question).

Therefore, it is one-one function.

It is not onto function all values of the co-domain do not come out as output as 0 doesn’t come out as output.

For the inverse of the function, input is the set of values is Range and output is the set of values in domain. But in this case, the value 0 isn’t mapped with any element. So, function is not invertible.

In other words, a function has to be both one-one and onto for it to be invertible.

So, function isn’t invertible.

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(b)

f(x,y,z) = (z,y,x)

Here x can be a or b, y can be a or b and z can be a or b.

So, the possible inputs and outputs of the function are the following:

X

Y

Z

ZYX

1

A

A

A

AAA

2

A

A

B

BAA

3

A

B

A

ABA

4

A

B

B

BBA

5

B

A

A

AAB

6

B

A

B

BAB

7

B

B

A

ABB

8

B

B

B

BBB

So, every input has a distinct output. So, function is one-one.

And every element of co-domain is part of the range. So, function is onto.

Since, it is both one-one and onto. Therefore it is invertible also.

In the original function, the output involves swapping the first value with the third value in order to get the output. Notice that in order to use the output and determine the input, the same method must be followed - swap the first and third values.

Therefore, the inverse function is f(x,y,z) = (z,y,x). This is same as the original function.

Also notice, that the output is the reverse of input. So how do you get the input back? By reversing again. Therefore, the original function also serves as the inverse function.

X	1	2	3	4	5	6	7	8
F(X)	1	2	3	4	5	6	7	8