Let f:[0,3] A be defined by f(x)=2 x^3-15 x^2+36 x+7 and g:[0, ) B be defined by g(x)=x^20…

Question

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

$$\text { now } \begin{aligned}
f^{\prime}(x) & =6 x^2-30 x+36 \
& =6(x-2)(x-3)
\end{aligned}$$

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

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

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

also for range of $g(x)$

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

Solutions · 1