Question

If f : R -> A, given by f(x) = x2– 2x + 2 is onto function, find set A.

Answer 1

Answer:

Explanation:

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Answer 2

The Set A is =  [ 1, ∞ )

Given: f : R -> A and f(x) = x² – 2x + 2 and f(x) is onto function.

To Find: Set A.

Solution:

1. The major property of onto function is that for such a function,

           Range of the function = Co-domain.

2. Range can be found by calculating the minimum and maximum values of the function.

3. In the form, f: R -> A, R is the domain of the function and A is its co-domain. So, we need to find the co-domain itself.

We have f(x) = x² – 2x + 2

               f(x) = x² – 2x + 1 + 1

               f(x) = ( x – 1 )² + 1                                         ...... (1)

From (1), we can see that the function f(x) becomes minimum when (x-1)² becomes zero. Thus, at x = 1, f(x) is minimum which is 1.

               f(x) = (1 - 1)² + 1 = 1

Now, we must find the maximum value that the function attend

Since f(x) has a square term so, it will always be positive, so the maximum value of f(x) is infinity (∞).

So Range = [ 1 , ∞ )

We already that for the Onto function, co-domain (A) = Range

Thus, A = [ 1 , ∞ ).

Hence, the set A is =  [ 1, ∞ )

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