Sets, Relations & Functions
Range of Functions / Integer Points
nta_pyq_2025_apr
Grade 11
Question:
Let $f : [0,3] \to A$ be defined by $f(x) = 2x^3 - 15x^2 + 36x + 7$ and $g : [0, \infty) \to B$ be defined by $g(x) = \dfrac{x^{2025}}{x^{2025}+1}$. If both functions are onto and $S = \{x \in \mathbb{Z} : x \in A \text{ or } x \in B\}$, then $n(S)$ is equal to:
Step-by-Step Solution
Key Concept: Find range A of $f$ on $[0,3]$ by checking critical points and endpoints. Find range B of $g$ on $[0,\infty)$. Then count integer elements in $A\cup B$.
$f'(x)=6(x-2)(x-3)$. $f(0)=7,f(2)=35,f(3)=34$. Range $A=[7,35]$. $g$ is increasing, $g(0)=0$, $g\to1$: Range $B=[0,1)$. $A\cup B$: integers in $[0,1)\cup[7,35]$ = $\{0\}\cup\{7,8,\ldots,35\}$ = 1+29 = 30 integers.
Correct Answer: 30