Q57 / 150 · MHT CET 2026 (16 April Shift 2) · Mathematics
If $e^x + e^{f(x)} = e$, then the domain of $f(x)$ is
- A$(1, \infty)$
- Correct answerB$(-\infty, 1)$
- C$(-\infty, \infty)$
- D$(-\infty, 0)$
Solution
Given $e^x + e^{f(x)} = e$
$e^{f(x)} = e - e^x$
Taking natural logarithm on both sides, we get
$f(x) = \ln(e - e^x)$
For $f(x)$ to be defined, the argument of the logarithm must be strictly positive.
$e - e^x > 0$
$e^x < e$
$x < 1$
Therefore, the domain of $f(x)$ is $(-\infty, 1)$.
Answer: $(-\infty, 1)$
$e^{f(x)} = e - e^x$
Taking natural logarithm on both sides, we get
$f(x) = \ln(e - e^x)$
For $f(x)$ to be defined, the argument of the logarithm must be strictly positive.
$e - e^x > 0$
$e^x < e$
$x < 1$
Therefore, the domain of $f(x)$ is $(-\infty, 1)$.
Answer: $(-\infty, 1)$
Hence, the correct answer is option: 'B'.