Set, Relation and FunctionMedium
Question
The function f : [0, 3] → [1, 29], defined by f(x) = 2x3 - 15x2 + 36x + 1, is
Options
A.one-one and onto
B.onto but not one-one
C.one-one but not onto
D.neither one-one nor onto
Solution
f (x) = 2x3 - 15x2 + 36x + 1
f′(x) = 6x2 - 30x + 36
= 6 (x2 - 5x + 6)
= 6 (x - 2) (x - 3)
f (x) is increasing in [0, 2] and decreasing in [2, 3]
f (x) is many one
f (0) = 1
f (2) = 29
f (3) = 28
Range is [1, 29]
Hence, f (x) is many-one-onto
f′(x) = 6x2 - 30x + 36
= 6 (x2 - 5x + 6)
= 6 (x - 2) (x - 3)
f (x) is increasing in [0, 2] and decreasing in [2, 3]
f (x) is many one
f (0) = 1
f (2) = 29
f (3) = 28
Range is [1, 29]
Hence, f (x) is many-one-onto
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