Find the domain and the inverse of the following functions:
$f (x) = \frac{1}{1 - x^{2}}$

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Solution

The function $f (x) = \frac{1}{1 + x^{2}}$ is only defined when $1 + x^{2} \neq 0 x^{2} \neq - 1 x \in R$
Let $y = \frac{1}{1 - x^{2}} 1 - x^{2} = \frac{1}{y} x^{2} = 1 - \frac{1}{y} x = \sqrt{\frac{y - 1}{y}} f^{- 1} (x) = \sqrt{\frac{y - 1}{y}}$

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Find the domain and the inverse of the following functions:
$f (x) = \frac{1}{1 - x^{2}}$

The function $f (x) = \frac{1}{1 + x^{2}}$ is only defined when $1 + x^{2} \neq 0 x^{2} \neq - 1 x \in R$
Let $y = \frac{1}{1 - x^{2}} 1 - x^{2} = \frac{1}{y} x^{2} = 1 - \frac{1}{y} x = \sqrt{\frac{y - 1}{y}} f^{- 1} (x) = \sqrt{\frac{y - 1}{y}}$