First slide

Functions (XII)

Question

The function $f : [- 1 / 2, 1 / 2] \to [- π / 2, π / 2]$ defined by $f (x) = \sin^{- 1} (3 x - 4 x^{3})$ is

Moderate

A

both one-one and onto
B

neither one-one nor onto
C

onto but not one-one
D

one-one but not onto

Solution

Since $\sin^{- 1} (3 x - 4 x^{3}) = 3 \sin^{- 1} x \in [- π / 2, π / 2]$
i.e. $\sin^{- 1} x \in [- π / 6, π / 6]$ or $x \in [- 1 / 2, 1 / 2]$ so $f$ is onto
Also $f^{'} (x) = \frac{3}{\sqrt{1 - x^{2}}} > 0$ for $- 1 / 2 < x < 1 / 2$
Therefore, $f$ increases on $[- 1 / 2, 1 / 2]$ and hence f is one-one.

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