CUET · MATHS · PYQ PAPER 2023
If a function \(f: R _{+} \rightarrow[-5, \infty)\) given by \(f(x)=9 x^2+6 x-5\) is one-one and onto, then \(f^{-1}(y)\) is
- A \(\frac{\sqrt{y+1}+6}{3}\)
- B \(\frac{\sqrt{y+5}+1}{3}\)
- C \(\frac{\sqrt{y+6}+1}{3}\)
- D \(\frac{\sqrt{y+6}-1}{3}\)
Answer & Solution
Correct Answer
(D) \(\frac{\sqrt{y+6}-1}{3}\)
Step-by-step Solution
Detailed explanation
\(y = 9x^2+6x-5\) \(y = (3x)^2+2(3x)(1)+1^2-1^2-5\) \(y = (3x+1)^2-6\) \(y+6 = (3x+1)^2\) \(\sqrt{y+6} = 3x+1\) \(\sqrt{y+6}-1 = 3x\) \(x = \frac{\sqrt{y+6}-1}{3}\) \(f^{-1}(y) = \frac{\sqrt{y+6}-1}{3}\)
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