Function

Algebra Level 4

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

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one-one and onto neither one-one nor onto one-one but not onto onto but not one-one

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Tom Engelsman
Apr 3, 2016

For a function f(x) to be one-to-one (or injective), each x in the domain of f yields only one unique value in its range. And for f(x) to be deemed as onto (or surjective), all of its corresponding range values must be included over its given domain.

Sketching f(x) = 2x^3 - 15x^2 + 36x + 1 over the domain [0,3] yields f(0) = 1, a maximum at x = 2 (or f(2) = 29) and f(3) = 28. One can easily use calculus to verify these points out.

For [0,3] => [1,29], f(x) is clearly onto. However, it is not one-to-one since there exist repeated range values over the interval (2,3].

Function

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