 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.

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.

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.