First slide

Functions (XII)

Question

Let f(x) = (x + 1)² – 1, x ≥ –1 then the set {x : f(x) = f ^–1(x)} is equal to

Moderate

A

{0, – 1}
B

{0, 1}
C

{– 1, 1}
D

{0}

Solution

Let y = (x + 1)² – 1, x ≥ – 1
$\Rightarrow$ (x + 1)² = y + 1
$\Rightarrow$ x + 1 = y + 1 as x ≥ – 1
$\Rightarrow$ $x = - 1 + \sqrt{y + 1}, y \geq - 1$
Thus $f^{- 1} (x) = - 1 + \sqrt{x + 1}$
So f (x) = f ^–1(x)
$\Rightarrow {(x + 1)}^{2} - 1 = - 1 + \sqrt{x + 1}$
$\Rightarrow \sqrt{x + 1} = 0 o r {(x + 1)}^{3 / 2} = 1$
$\Rightarrow$ x = – 1 or x = 0

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