Let $f=left{left(x, rac{x^{2}}{1+x^{2}} ight): x in mathbf{R} ight}$ be a function from $mathbf{R}$ into $mathbf{R}$. Determine the range of $f$.
Question:

Let $f=\left\{\left(x, \frac{x^{2}}{1+x^{2}}\right): x \in \mathbf{R}\right\}$ be a function from $\mathbf{R}$ into $\mathbf{R}$. Determine the range of $f$.

Solution:

$f=\left\{\left(x, \frac{x^{2}}{1+x^{2}}\right): x \in \mathbf{R}\right\}$

$=\left\{(0,0),\left(\pm 0.5, \frac{1}{5}\right),\left(\pm 1, \frac{1}{2}\right),\left(\pm 1.5, \frac{9}{13}\right),\left(\pm 2, \frac{4}{5}\right),\left(3, \frac{9}{10}\right),\left(4, \frac{16}{17}\right), \ldots\right\}$

The range of f is the set of all second elements. It can be observed that all these elements are greater than or equal to 0 but less than 1.

[Denominator is greater numerator]

Thus, range of f = [0, 1)

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