First slide

Functions (XII)

Question

Let f be a function defined by
$f (x) = (x - 1)^{2} + 1, (x \geq 1)$
Statement-1: The set {x : f(x) = f ^–1(x)} = {1, 2}
Statement-2: $f$ is a bijection and $f^{- 1} (x) = 1 + \sqrt{x - 1}, x \geq 1$

Moderate

A

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
B

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
C

STATEMENT-1 is True, STATEMENT-2 is False
D

STATEMENT-1 is False, STATEMENT-2 is True

Solution

Let y = f(x) = (x – 1)²+ 1
$y - 1 = {(x - 1)}^{2} \Rightarrow x = 1 + \sqrt{y - 1}, y \geq 1$
Thus $f^{- 1} (x) = 1 + \sqrt{x - 1}, x \geq 1$ . So statement-2 is true
Now f(x) = f ^–1(x)
$\Rightarrow$ ${(x - 1)}^{2} = \sqrt{x - 1}$
$\Rightarrow$ $\begin{array}{l} \sqrt{x - 1} [(x - 1)^{3 / 2} - 1] = 0 \end{array}$
$\Rightarrow x = 1, 2$
So statement-1 is true and statement-2 is a correct explanation for statement-1.

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