Question:
If \( e^x = y + \sqrt{1 + y^2} \), then the value of \( y \) is
If \( e^x = y + \sqrt{1 + y^2} \), then the value of \( y \) is
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Use algebraic manipulation to isolate the desired variable in exponential equations.
Updated On: Jan 12, 2026
- \( \frac{1}{2} (e^x - e^{-x}) \)
- \( \frac{1}{2} (e^x + e^{-x}) \)
- \( e^x \)
- \( e^x + 2 \)
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The Correct Option is A
Solution and Explanation
Step 1: Isolate y.
Rearranging the given equation \( e^x = y + \sqrt{1 + y^2} \), we can solve for \( y \). The solution simplifies to \( y = \frac{1}{2} (e^x - e^{-x}) \).
Step 2: Conclusion.
The correct answer is (A), \( \frac{1}{2} (e^x - e^{-x}) \).
Rearranging the given equation \( e^x = y + \sqrt{1 + y^2} \), we can solve for \( y \). The solution simplifies to \( y = \frac{1}{2} (e^x - e^{-x}) \).
Step 2: Conclusion.
The correct answer is (A), \( \frac{1}{2} (e^x - e^{-x}) \).
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