First slide

Functions (XII)

Question

Which of the following functions have inverse defined on their ranges ?

Difficult

A

$f (x) = x^{2}, x \in R$
B

$f (x) = x^{3}, x \in R$
C

$f (x) = \sin x, 0 < x < 2 π$
D

$f (x) = e^{x}, x \in R$

Solution

The function in (a) is not one-one, so $f^{- 1}$ is not defined.
The function in (b) is one-one and onto defined from $R \to R,$ hence $f^{- 1}$ is defined from $R \to R$ given by $f^{- 1} (x) = x^{1 / 3} .$

The function $f (x) = \sin x$ is not one-one in $(0, 2 π)$ as $f (π / 4) = f (3 π / 4) = 1 / \sqrt{2},$ so it is not invertible.

The function $f (x) = e^{x}$ is one-one as $e^{x} = e^{y} \Leftrightarrow x = y .$

The range $f = R^{+} = {x : x > 0} .$

So inverse of $f$ $i e \log x$ is defined on $R^{+}$ to $R .$

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