Suppose f : R → R is defined by \({\rm{f}}\left( {\rm{x}} \right) = \frac{{{{\rm{x}}^2}}}{{1 + {{\rm{x}}^2}}}\) What is the range of the function?

Concept:

Below are steps to find out range of function:

1. Write down y = f(x) and then solve the equation for x, giving something of the form x=g(y).

Find the domain of g(y), and this will be the range of f(x).

If you can't seem to solve for x, then try graphing the function to find the range.

Calculation:

Given that,

\({\rm{f}}\left( {\rm{x}} \right) = \frac{{{{\rm{x}}^2}}}{{1 + {{\rm{x}}^2}}}\)

Put f(x) = y

\({\rm{y}} = \frac{{{{\rm{x}}^2}}}{{1 + {{\rm{x}}^2}}}\)

Cross multiply

⇒ y(1 + x² ) = x²

⇒ y + y . x² = x²

⇒ x² (1 -y) = y

⇒ x² = y/(1 -y)

⇒ \({\rm{x}} = \sqrt {\frac{{\rm{y}}}{{1 - {\rm{y}}}}}\)

⇒ x = g(y) = \(\sqrt {\frac{{\rm{y}}}{{1 - {\rm{y}}}}} \)

Now we find out domain of g(y) which will be range of f(x)

So as we know inside root always positive values comes so,

⇒ y ≠ 1 and \(\frac{{\rm{y}}}{{{\rm{y}} - 1}} > 0\)