Question:
Let \( f(x) = x^2 - 10x - 19, \, x \in \mathbb{R} \). Then the inverse image of 5, \( f^{-1}(5) = \)
Let \( f(x) = x^2 - 10x - 19, \, x \in \mathbb{R} \). Then the inverse image of 5, \( f^{-1}(5) = \)
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Inverse image means solving \( f(x)=k \), not finding inverse function.
Updated On: Apr 21, 2026
- \( \{-2, -12\} \)
- \( \{-2, 12\} \)
- \( \{2, -12\} \)
- \( \{2, 12\} \)
- \( \phi \)
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The Correct Option is B
Solution and Explanation
Concept:
Inverse image means solving:
\[
f(x) = 5
\]
Step 1: Set equation.
\[ x^2 - 10x - 19 = 5 \]
Step 2: Simplify.
\[ x^2 - 10x - 24 = 0 \]
Step 3: Factorize.
\[ x^2 - 10x - 24 = (x-12)(x+2)=0 \]
Step 4: Solve.
\[ x = 12, -2 \] \[ f^{-1}(5)=\{-2,12\} \]
Step 1: Set equation.
\[ x^2 - 10x - 19 = 5 \]
Step 2: Simplify.
\[ x^2 - 10x - 24 = 0 \]
Step 3: Factorize.
\[ x^2 - 10x - 24 = (x-12)(x+2)=0 \]
Step 4: Solve.
\[ x = 12, -2 \] \[ f^{-1}(5)=\{-2,12\} \]
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