The function f:[0, ) R given by f(x)= x x+1 is: - Vidyadip

MCQ

The function $\text{f}:[0,\infty)\rightarrow\ \text{R}$ given by $\text{f(x)}=\frac{\text{x}}{\text{x}+1}$ is:

A
One$-$one and onto.
✓
One$-$one but not onto.
C
Onto but not one$-$one.
D
Onto but not one$-$one.

✓

Answer

Correct option: B.

One$-$one but not onto.

Given function is $\text{f(x)}=\frac{\text{x}}{\text{x}+1}$ on $\text{f}:[0,\infty)\rightarrow\ \text{R}$
If $f(x) = f(y)$
$\Rightarrow\ \frac{\text{x}}{\text{x}+1}=\frac{\text{y}}{\text{y}+1}$
$\Rightarrow xy + x = xy + y$
$\Rightarrow x = y$
Hence$, f$ is one$-$one.
If $y = f(x)$
$\text{y}=\frac{\text{x}}{\text{x}+1}$
$\Rightarrow xy + y = x$
$\Rightarrow xy - x = -y$
$x(y - 1) = -y$
$\text{x}=\frac{-\text{y}}{\text{y}-1}\neq\text{f(x)}$
It is not onto.

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