The function mathrmf:[ mathrma, ∞) arrow mathrmR where mathrmR denotes the range corresponding to the given domain, with rule f(x)=2 x3-3 x2+6, will have an inverse provided

Question

The function mathrmf:[ mathrma, ∞) arrow mathrmR where mathrmR denotes the range corresponding to the given domain, with rule f(x)=2 x3-3 x2+6, will have an inverse provided (A)  mathrma ≥ 1 (B)  mathrma ≥ 0 (C)  mathrma ≤ 0 (D)  mathrma ≤ 1

Tardigrade Admin · Accepted Answer

mathrmf( mathrmx)=2 mathrmx3-3 mathrmx2+6 f prime(x)=6 x2-6 x=6(x2-x)=0 gives x=0 or x=1 for inverse to exist function must be one one onto hence Domain is [1, ∞) Hence a ≥ 1

Q. The function $f : [a, \infty) \to R$ where $R$ denotes the range corresponding to the given domain, with rule $f (x) = 2 x^{3} - 3 x^{2} + 6$ , will have an inverse provided

$a \geq 1$

$a \geq 0$

$a \leq 0$

$a \leq 1$

$f (x) = 2 x^{3} - 3 x^{2} + 6$
$f^{'} (x) = 6 x^{2} - 6 x = 6 (x^{2} - x) = 0$ gives $x = 0$ or $x = 1$
for inverse to exist function must be one one onto
hence Domain is $[1, \infty)$
Hence $a \geq 1$

Q. The function f:[a,∞)→R where R denotes the range corresponding to the given domain, with rule f(x)=2x3−3x2+6, will have an inverse provided

a≥1

a≥0

a≤0

a≤1