Question

Let \[f = \left\{ {\dfrac{{{x^2}}}{{1 + {x^2}}}:x \in R} \right\}\] be a function from R to R. Determine the range of f.

Answer 1

Hint: The domain of a function is the set of all values for which the function is defined. The range of the function is the set of all values that f outputs. Use the definition to find the range of the given function.

Complete step-by-step answer:

A function is a relation from a set of inputs to a set of possible outputs where each input is related to one and only one output.

A function defined on all real numbers is called a real function.

We are given the function \[f = \left\{ {\dfrac{{{x^2}}}{{1 + {x^2}}}:x \in R} \right\}\] defined on real numbers, hence, it is a real function.

The domain of a function is the set of all values for which the function is defined. The range of the function is the set of all values that f outputs.

We need to find the range of \[f = \left\{ {\dfrac{{{x^2}}}{{1 + {x^2}}}:x \in R} \right\}\].

For all real number x, we have:

\[1 > 0\]

Adding \[{x^2}\] to both the sides, we have:

\[{x^2} + 1 > {x^2} \geqslant 0\]

Now we divide all the three terms by \[{x^2} + 1\] which is a positive number, hence, the inequality is retained.

\[1 > \dfrac{{{x^2}}}{{{x^2} + 1}} \geqslant 0\]

Hence, we have the range of the function as \[[0,1)\].

Note: You can also find the range of the number by substituting the values in the domain and see what is the range of the results you get.