Let f:[0,3] A be defined by f(x)=2x^3-15x^2+36x+7 and g:[0,) B be defined by g(x)=x^2025x^…

Question

Let $f:[0,3]	o$ A be defined by $f(x)=2{x}^{3}-15{x}^{2}+36x+7$ and $g:[0,\infty)	o B$ be defined by $g(x)=\frac{{x}^{2025}}{{x}^{2025}+1}$, If both the functions are onto and $S={x\in Z;x\in A$ or $x\in B}$, then $n(S)$ is equal to :

Accepted Answer

as $f(x)$ is onto hence $A$ is range of $f(x)$

$\begin{gathered} {f}^{'}(x) =6{x}^{2}-30x+36 \ =6(x-2)(x-3) \end{gathered}$

$f(2)=16-60+72+7=35$

$\begin{gathered} f(3)=54-135+108+7=34 \ f(0)=7 \end{gathered}$

hence range $\in [7,35]=A$

also for range of $g(x)$

$\begin{gathered} g(x)=1-\frac{1}{{x}^{2025}+1}\in [0,1)=B \ s={0,7,8,\ldots ..35}n(s)=30 \end{gathered}$

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