ISRO-DEC2017-9

Question

The function $f:[0,3]\rightarrow [1,29]$ defined by $f(x)=2x^{3}-15x^{2}+36x+1$ is

• A. injective and surjective
• B. surjective but not injective
• C. injective but not surjective
• D. neither injective nor surjective

Comments

minimum means graph decreasing

right?

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@srestha,

no,  local minimum(or simply minimum) is a point where $f'(x)=0\ and\ f''(x)>0$ or simply we can say that local minimum is a point from where f(x) starts to increase on increasing x.

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@joshi_nitish

from definition, i think it is injective but not surjective, as every element in [1,29] is not mapped and there is no similar values fo r x in y for x -> y mapping.

Answer 1

Injective means for every x there is unique value of f(x)

So here for x= 0 we have f(x) = 1

                  x= 1   f(x)= 24

                  x= 2   f(x)= 29

                  x= 2   f(x)= 28

So,every x has unique value of f(x)

Thus,it should be injective

Now coming to surjective part..if every element of set B(HERE [1,29]) is image of set A(here [0,3]) tehn it will be surjective but it is not the case.

So, answer should be C

But, according to below link answer is B

Please check if my logic or understanding is wrong

https://books.google.co.in/books?id=_RAwDwAAQBAJ&pg=PA32&lpg=PA32&dq=the%20function%20f%3A[0%2C3]%20-%3E%20[1%2C29]%20defined&source=bl&ots=2g8axUMuzW&sig=EZzVx5E7L-uP6rXmRZaLUXc4Lig&hl=en&sa=X&ved=0ahUKEwjz48Puw5rYAhXJsI8KHYgED3YQ6AEISTAF#v=onepage&q&f=false

Comments

every value (28,29) is mapped by two values of x - how?

f((0)=1

f(1)=24

f(2)=29

f(3)=28

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see nitish comment

Here once function increasing, then decreasing, then again will increase

When the function will increase again?

When there is a local minima

that are of point 3 and 2

So, it cannot be one to one

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oops..i was checking for only the discrete values of x

Answer 2

one to one (injective)----->A function for which every element of the range of the function corresponds to exactly one element of the domain.

onto (surjective)------>The function is surjective (onto) if each element of the range is mapped to by at least one element of the domain.

Comments

why you're only checking for discrete values?

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It is not one to one