First slide

Functions (XII)

Question

Let $f (x) = {(x + 1)}^{2} - 1 (x \geq - 1) .$ Then the set
$S = {x : f (x) = f^{- 1} (x)}$ contains

Moderate

A

$\{0, - 1, \frac{- 3 + i \sqrt{3}}{2}, \frac{- 3 - i \sqrt{3}}{2}\}$
B

${0, 1, - 1}$
C

${0, - 1}$
D

none of these

Solution

$\begin{array}{l} f (x) = f^{- 1} (x) \Rightarrow f (f (x)) = x \\ \Rightarrow {[(x + 1)^{2} - 1 + 1]}^{2} - 1 = x . \end{array}$
$\Rightarrow (x + 1)^{2} = x + 1 \Rightarrow x = 0$ or $x = - 1$
So $S = {0, - 1} .$

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