Relations and Functions
NCERT Exemplar Class 12
CBSE
Grade 12
Question:
Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = x^2 + 3$. Find (i) $f^{-1}(28)$, (ii) $f^{-1}(2)$, (iii) $f^{-1}([4, 7])$.
Step-by-Step Solution
(i) $x^2+3=28 \Rightarrow x^2=25 \Rightarrow x = \pm 5 \Rightarrow f^{-1}(28) = \{-5, 5\}$. [1.5 Marks]
(ii) $x^2+3=2 \Rightarrow x^2=-1 \Rightarrow f^{-1}(2) = \emptyset$. [1.5 Marks]
(iii) $1 \le x^2 \le 4 \Rightarrow x \in [-2, -1] \cup [1, 2]$. [2.0 Marks]
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🎯 Official CBSE Marking Scheme:
Evaluating $f^{-1}(28) = \{-5, 5\}$: 1.5 Marks
Evaluating $f^{-1}(2) = \emptyset$: 1.5 Marks
Evaluating $f^{-1}([4, 7]) = [-2, -1] \cup [1, 2]$: 2.0 Marks
Correct Answer:
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