Question

How to find the range of $\dfrac{{{x^2}}}{{1 - {x^2}}}$ ?

Answer 1

We are given an expression and asked to find the range of the given expression. The expression is in terms of $x$ so first try to find the value of $x$. Then observe the condition you obtain and check for the set of possible values. This will give you the range for the given expression.

Complete step by step solution:
Given, the expression $\dfrac{{{x^2}}}{{1 - {x^2}}}$.
We are asked to find the range of the given expression.
Let $y = \dfrac{{{x^2}}}{{1 - {x^2}}}$ and try to solve for $x$.
We have,
$y = \dfrac{{{x^2}}}{{1 - {x^2}}}$
$\Rightarrow y = \dfrac{{1 - \left( {1 - {x^2}} \right)}}{{1 - {x^2}}}$
$\Rightarrow y = \dfrac{1}{{1 - {x^2}}} - \dfrac{{\left( {1 - {x^2}} \right)}}{{1 - {x^2}}}$
$\Rightarrow y = \dfrac{1}{{1 - {x^2}}} - 1$
$\Rightarrow y + 1 = \dfrac{1}{{1 - {x^2}}}$
Multiplying both sides of the equation by $1 - {x^2}$ we get,
$\left( {1 - {x^2}} \right)\left( {y + 1} \right) = \dfrac{{\left( {1 - {x^2}} \right)}}{{1 - {x^2}}}$
$\Rightarrow \left( {1 - {x^2}} \right)\left( {y + 1} \right) = 1$
Dividing both sides of the equation by $\left( {y + 1} \right)$ we get,
$\dfrac{{\left( {1 - {x^2}} \right)\left( {y + 1} \right)}}{{y + 1}} = \dfrac{1}{{y + 1}}$
$\Rightarrow \left( {1 - {x^2}} \right) = \dfrac{1}{{y + 1}}$
$\Rightarrow - {x^2} = \dfrac{1}{{y + 1}} - 1$
$\Rightarrow {x^2} = 1 - \dfrac{1}{{\left( {y + 1} \right)}}$
$\Rightarrow {x^2} = \dfrac{{y + 1 - 1}}{{y + 1}}$
$\Rightarrow {x^2} = \dfrac{y}{{y + 1}}$
For the solution to the real the term $\dfrac{y}{{y + 1}}$ must be equal to or greater than zero.
That is,
$\dfrac{y}{{y + 1}} \geqslant 0$
For the above condition, there can be two situations.
1. $y \geqslant 0$ and $y + 1 > 0$ which implies $y \in \left[ {0,\infty } \right)$.
2. $y < 0$ and $y + 1 < 0$ which implies $y \in \left( { - \infty , - 1} \right)$.

Therefore, the range of $f(x) = \dfrac{{{x^2}}}{{1 - {x^2}}}$ is $y \in \left( { - \infty , - 1} \right) \cup \left[ {0,\infty } \right)$.

Note: Range of a function can be defined as the set of all possible values for the output of the function. There is another term called domain. Domain of a function is the set of all possible values for the input of the function. While finding the domain and range of a function, remember a few points. First, the denominator of the function cannot be zero as it will make the function infinite and secondly if there is a square root then the number under the square root must be a positive number as if we take the number to be negative the result will be a complex number.