First slide

Functions (XII)

Question

The range of the function $y = \sin^{- 1} (\frac{x^{2}}{1 + x^{2}}) is$

Moderate

A

$(0, \frac{π}{2})$
B

$[0, \frac{π}{2})$
C

$[0, \frac{π}{2}]$
D

None of these

Solution

Clearly, for y to be defined, $|\frac{x^{2}}{1 + x^{2}}| \leq 1$ which is true for all x ∈ R. So, the domain = (– ∞, ∞).
$Now, y = \sin^{- 1} (\frac{x^{2}}{1 + x^{2}}) \Rightarrow \frac{x^{2}}{1 + x^{2}} = \sin y \Rightarrow x = \frac{x^{2}}{1 + x^{2}}$
For x to be real, sin y ≥ 0 and 1 – sin y > 0
$\begin{array}{l} \Rightarrow 0 \leq \sin y < 1 \Rightarrow y \in [0, \frac{π}{2}) \\ ∴ Range = [0, \frac{π}{2}) \end{array}$

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