JEE Main 2012 Mathematics question on Sets Relations and Functions
The function f:[0,3]→[1,29], defined by f(x)=2x3−15x2+36x+1, is
A.
one-one and onto
B.
onto but not one-one
C.
one-one but not onto
D.
neither one-one nor onto
Show answer and solution
Correct option: B
Solution
f'(x) = 6x² - 30x + 36 = 6(x-2)(x-3), so f has a local max at x=2 and local min at x=3 on [0,3]. Computing: f(0)=1, f(2)=29, f(3)=28. Since f increases then decreases, it is not one-one (e.g. f(0)=1 and f(3)=28, but values near 1 are hit twice). The range is [1,29] (minimum value 1 at x=0, maximum 29 at x=2), which exactly equals the codomain [1,29], so f is onto.
[AIEEE 2012]
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