First slide

Functions (XII)

Question

$The function f : (- \infty, - 1) \to (0, e^{5}] defined by f (x) = e^{x^{3} - 3 x + 2} is$

Moderate

A

many-one and onto
B

many-one and into
C

one-one and onto
D

one-one and into

Solution

$f (x) = e^{x^{3} - 3 x + 2} \Rightarrow f' (x) = e^{x^{3} - 3 x + 2} \cdot (3 x^{2} - 3) > 0 f o r a l l x \in (- \infty, - 1)$
$t h e r e f o r e f i s i n c r e a \sin g f u n c t i o n a n d h e n c e i t i s o n e - o n e$
$R a n g e o f f (x) = (0, e^{4}) (\sin c e x t e n d s t o - \infty \Rightarrow f (x) t e n d s t o 0 a n d x \to - 1 \Rightarrow f (x) \to e^{4})$
$R a n g e o f f \neq C o d o m a i n o f f \Rightarrow i t i s n o t o n t o (i . e i n t o)$

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