First slide

Functions (XII)

Question

$If f : [0, \infty) \to [0, \infty) and f (x) = \frac{x}{1 + x}, then f is$

Moderate

A

one-one and onto
B

one-one but not onto
C

onto but not one-one
D

neither one-one nor onto

Solution

$f (x) = \frac{x}{1 + x}$
$f' (x) = \frac{1 + x - x}{{(1 + x)}^{2}} = \frac{1}{{(1 + x)}^{2}} > 0 f o r a l l x \in R$
$t h e r e f o r e f (x) i s i n c r e a \sin g f u n c t i o n, h e n c e f i s o n e - o n e$
$d o m a i n o f f (x) i s x \in [0, \infty) \sin c e x < x + 1 \Rightarrow \frac{x}{x + 1} < 1$
$y = \frac{x}{x + 1} \in [0, 1)$ $\neq C o d o m a i n [0, \infty)$
so it is not onto

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