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^(2025)/x^(2025) +1.

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 \(S=\{x \in \mathbf{Z}: x \in A\) or \(x \in B\},\) then n(S) is equal to :

(1) 30

(2) 36

(3) 29

(4) 31

Answer 1

Correct option is (1) 30 

As f(x) is onto hence A is range of f(x)

now \(f^{\prime}(x)=6 x^{2}-30 x+36\)

\(=6(x-2)(x-3) \)

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

\(\mathrm{f}(3)=54-135+108+7=34\)

f(0) = 7

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

also for range of g(x)

\(g(x)=1-\frac{1}{x^{2025}+1} \in[0,1)=B\)

\(\mathrm{s}=\{0,7,8, \ldots . .35\}\) hence \(\mathrm{n}(\mathrm{s})=30\)