cmi-entrance 2025 Q13

cmi-entrance · India · pgmath 10 marks Sequences and Series Functional Equations and Identities via Series

Let $f , g , h$ be functions from $\mathbb { R }$ to $\mathbb { R }$ such that $$h ( f ( x ) + g ( y ) ) = x y$$ for all $x , y \in \mathbb { R }$. Show the following:
(A) $(2$ marks$)$ $h$ is surjective.
(B) $(3$ marks$)$ If $f$ is continuous then $f$ is strictly monotone.
(C) $(5$ marks$)$ There do not exist continuous functions $f , g , h$ satisfying $(*)$.

Let $f , g , h$ be functions from $\mathbb { R }$ to $\mathbb { R }$ such that
$$h ( f ( x ) + g ( y ) ) = x y$$
for all $x , y \in \mathbb { R }$. Show the following:\\
(A) $(2$ marks$)$ $h$ is surjective.\\
(B) $(3$ marks$)$ If $f$ is continuous then $f$ is strictly monotone.\\
(C) $(5$ marks$)$ There do not exist continuous functions $f , g , h$ satisfying $(*)$.

Paper Questions

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