If $f(x)=2{x}^{3}-15{x}^{2}+36x+7:[0,3]\to A$ $g(x)=\frac{{x}^{2025}}{1+{x}^{2025}}:[0,\infty )\to B$ $f(x)$ and g(x) are onto functions. $S={x∣x\in Z,x\in A$or $x\in B}$. Find n(S).

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YESPYQ · Accepted Answer

Correct answer: c) 30. ${f}^{'}(x)=6{x}^{2}-30x+36=6\left({x}^{2}-5x+6\right)$ $=6(x-2)(x-3)$ $f(0)=7\ f(2)=16-60+72+7=35\ f(3)=54-135+108+7=34\ ∴A=\left[7,35\right]\ \mathrm{g}(\mathrm{x})=\frac{{\mathrm{x}}^{2025}}{1+{\mathrm{x}}^{2025}}=1-\frac{1}{1+{\mathrm{x}}^{2025}}\ {\mathrm{x}}^{2025}\geq 0\Rightarrow ∴1+{\mathrm{x}}^{2025}\geq 1\ \Rightarrow 0 1-\frac{1}{1+{\mathrm{x}}^{2025}}\geq 0∴\mathrm{B}=[0,1)\ ∴\mathrm{S}={0,7,8,\ldots \ldots ,35},n(S)=30$

If \(f(x)=2{x}^{3}-15{x}^{2}+36x+7:[0,3]\to A\) \(g(x)=\frac{{x}^{2025}}{1+{x}^{2025}}:[0,\infty )\to B\) \(f(x)\) and g…

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