First slide

Functions (XII)

Question

If the following functions are defined from [-1, 1] to [-1, 1], select those which are not bijective.

Moderate

A

$\begin{array}{l} \sin (\sin^{- 1} x) \end{array}$
B

$\begin{array}{l} \frac{1}{π} \sin^{- 1} (\sin x) \end{array}$
C

$(s g n (x)) \ln (e^{x})$
D

$x^{3} (s g n (x))$

Solution

$f (x) = \sin (\sin^{- 1} x) = x \forall x \in [- 1, 1],$ which is one-one and onto.
$f (x) = \frac{2}{π} \sin^{- 1} (\sin x) = \frac{2}{π} x$
The range of the function for
$x \in [1 -, 1] is [- \frac{2}{π}, \frac{2}{π}],$
which is a subset of [-1, 1].
Hence, the function is one-one but not onto and, hence, is not bijective.
$f (x) = (s g n (x)) \ln (e^{x}) = (\sin (x)) x = \{\begin{cases} x, x > 0 \\ - x, x < 0 \\ 0, x = 0 \end{cases}$
This function has range [0, 1] which is a subset of [-1, l].
Hence, the function is into. Also, the function is many-one.
$f (x) = x^{3} s g n (x) = \{\begin{cases} x, x > 0 \\ - x^{3}, x < 0 \\ 0, x = 0 \end{cases}$
which is many-one and into.

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