Let f: (1, 3) arrow R be a function defined by f (x)=(x[x]/1+x2), where [x] denotes the greatest integer le x. Then the range of f is :

Question

Let f: (1, 3) arrow R be a function defined by f (x)=(x[x]/1+x2), where [x] denotes the greatest integer le x. Then the range of f is : (A) ((2/5), (3/5))∪ ((3/4), (4/5)) (B) ((2/5), (4/5)) (C) ((3/5), (4/5)) (D) ((2/5), (1/2)) ∪ ((3/5), (4/5))

Tardigrade Admin · Accepted Answer

f(x) = begincases (x/x2+1) ; textx ∈(1, 2) [2ex] (2x/x2+1) ; textx ∈(2, 3) endcases f(x) is decreasing function ∴ f(x) ∈((2/5), (1/2)) ∪ ((3/5), (4/5))

Q. Let $f : (1, 3) \to R$ be a function defined by $f (x) = \frac{x [ x ]}{1 + x ^{2}},$ where $[x]$ denotes the greatest integer $\leq x .$ Then the range of $f$ is :

$(\frac{2}{5}, \frac{3}{5}) \cup (\frac{3}{4}, \frac{4}{5})$

$(\frac{2}{5}, \frac{4}{5})$

$(\frac{3}{5}, \frac{4}{5})$

$(\frac{2}{5}, \frac{1}{2}) \cup (\frac{3}{5}, \frac{4}{5})$

$f (x) = ⎩ ⎨ ⎧ \frac{x}{x ^{2} + 1}; \frac{2 x}{x ^{2} + 1}; x \in (1, 2) x \in (2, 3)$
$f (x)$ is decreasing function
$∴ f (x) \in (\frac{2}{5}, \frac{1}{2}) \cup (\frac{3}{5}, \frac{4}{5})$

Q. Let f:(1,3)→R be a function defined by f(x)=1+x2x[x]​, where [x] denotes the greatest integer ≤x. Then the range of f is :

(52​,53​)∪(43​,54​)

(52​,54​)

(53​,54​)

(52​,21​)∪(53​,54​)