Functions (Advanced) JEE Main PYQs, 2020 to 2025

If the range of the function $f(x) = \frac{5-x}{x^2 - 3x + 2} , \ x \neq 1, 2,$ is $(-\infty , \alpha] \cup [\beta, \infty)$, then $\alpha^2 + \beta^2$ is equal to :

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function satisfying $f(0)=1$ and $f(2 x)-f(x)=x$ for all $x \in \mathbb{R}$. If $\lim \limits_{n \rightarrow \infty}\left\{f(x)-f\left(\frac{x}{2^n}\right)\right\}=G(x)$, then $\sum\limits_{r=1}^{10} G\left(r^2\right)$ is equal to

Let $f(x)=x^5+2 x^3+3 x+1, x \in \mathbf{R}$, and $g(x)$ be a function such that $g(f(x))=x$ for all $x \in \mathbf{R}$. Then $\frac{g(7)}{g^{\prime}(7)}$ is equal to :

Let f(x) and g(x) be two real polynomials of degree 2 and 1 respectively. If $f(g(x)) = 8{x^2} - 2x$ and $g(f(x)) = 4{x^2} + 6x + 1$, then the value of $f(2) + g(2)$ is _________.