First slide

Functions (XII)

Question

If $f : R \to R$ is defined by $f (x) = x^{2} + 1$ then value of $f^{- 1} (17)$ and $f^{- 1} (- 3)$ are, respectively.

Moderate

A

$ϕ, {4, - 4}$
B

$ϕ, {3, - 3}$
C

${{3, - 3}, ϕ$
D

${4, - 4}, ϕ$

Solution

For any $A \subseteq R,$ we have
$\begin{array}{l} f^{- 1} (A) = {x \in R : f (x) \in A} . Thus, \\ f^{- 1} (17) = {x : f (x) \in {17}} = {x : f (x) = 17} \\ = \{x : x^{2} + 1 = 17\} = {4, - 4} . \end{array}$
and similarly, $∣ f^{- 1} (- 3) = \{x \in R : x^{2} + 1 = - 3\} = ϕ$

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