Suppose that $f : R \to R$ is a continuous function and $f (x) f (f (x)) = 1$ , for all real x. Let $f (10) = 5$ . Then for all such functions, which of the following statements is/are CORRECT?

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$f (x) = \frac{x}{4}$ has at least 3 solutions

$f (4) = \frac{1}{4}$

$f (x)$ can have at most of 9 integers in its range

$f (6) = \frac{1}{6}$

answer is A, C, D.

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Detailed Solution

$\Rightarrow P u t x = 10 f (10) + (5) = 1 \Rightarrow f (5) = \frac{1}{5} \Rightarrow f (x) = t \forall_{f (x)}^{l t} f (t) = \frac{1}{t} f (x) = \frac{1}{x} \forall x \in [\frac{1}{5}, 5]$

Study the graph by assuming for some $α$ ,

$f (α) = 10 (o r) > 10$

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