To find the domain of the function \( y = f(e^x) + f(\ln|x|) \), we need to ensure both \( f(e^x) \) and \( f(\ln|x|) \) are defined. Given that the domain of \( f(x) \) is \((0, 1)\), we need the following conditions:
- The expression \( e^x \) should be within the domain of \( f(x) \). Thus, \( 0 < e^x < 1 \), which implies:
- \( e^x < 1 \) leads to \( x < 0 \).
- \( e^x > 0 \) holds for all real \( x \), so no restriction here.
- The expression \( \ln|x| \) should be within the domain of \( f(x) \). Thus, \( 0 < \ln|x| < 1 \), which implies:
- \( \ln|x| > 0 \) gives \( |x| > 1 \), meaning \( x > 1 \) or \( x < -1 \).
- \( \ln|x| < 1 \) gives \( |x| < e \), meaning \(-e < x < e\).
Combining these two conditions for \( x \), we get:
- From both \( e^x < 1 \) (which gives \( x < 0 \)) and \( |x| > 1 \) (which requires \( x > 1 \) or \( x < -1 \)), the range for \( x \) is:
- The overlap from the conditions is \( x \in (\frac{1}{e}, 1) \).
Therefore, the domain of the expression \( y = f(e^x) + f(\ln|x|) \) is \(\left(\frac{1}{e}, 1\right)\).
Conclusion: The correct answer is \(\left(\frac{1}{e}, 1\right)\).
Similar Questions
For a real number $y$, consider $[y]$ denotes the greatest integer less than or equal to $y$.
If $f(x) = \frac{\tan(\pi[x-\pi])}{1+[x]^2}$, then- Given $P(x) = x^4 + ax^3 + bx^2 + cx + d$ such that $x=0$ is the only real root of $P'(x) = 0$. If $P(-1) < P(1)$, then in the interval $[-1, 1]$
- Consider a function $f(x)$ which has exactly two roots at $x=a$. If $\lim_{x \to a}\left(\frac{\lambda f'(x)}{f(x)} - \frac{1}{x-a}\right) = m \, (\neq 0)$, then the value of $\lambda$ is
- Which of the following statements is always true?
- Let $A = [a, \infty)$ denotes the domain, then $f: [a, \infty) \to B$, which is defined by $f(x) = 2x^3 - 3x^2 + 6$ will have an inverse for the smallest real value of '$a$' if
- Number of elements in the range set of $f(x) = \left[ \frac{x}{15} \right] \left[ -\frac{15}{x} \right]$, for all $x \in (0, 90)$; (where $[\cdot]$ denotes the greatest integer function) is
- Let domain and range of $f(x)$ and $g(x)$ is $[0, \infty)$. If $f(x)$ is an increasing function, $g(x)$ is a decreasing function, $h(x) = f\{g(x)\}, h(0) = 0$ and $p(x) = h(x^3 - 2x^2 + 2x) - h(4)$, then for all $x \in (0, 2)$
- A figure is bounded by the curves $y = x^2 + 1, y = 0, x = 0$ and $x = 1$. The point at which a tangent should be drawn to the curve $y = x^2 + 1$ for it to cut off trapezium of the greatest area from the figure is
- Let $f(x)$ be a twice differentiable function in $[1, 3]$ and $f(1) = f(3)$. Further if $|f''(x)| \le 2$, then for all $x$ in $[1, 3]$
- If $f$ be a real valued function defined for all real numbers $x$ such that for some fixed $a > 0$, it satisfies $f(x+a) = \frac{1}{2} + \sqrt{f(x) - (f(x))^2} \, \forall x$, then $f(x)$ is periodic with period
Important Questions from Differential Calculus
- Let $(2\alpha, \alpha)$ be the largest interval in which the function $f(t) = \frac{|t+1|}{t^2}, t < 0$, is strictly decreasing. Then the local maximum value of the function $g(x) = 2\log_e(x - 2) + \alpha x^2 + 4x - \alpha$, $x > 2$, is ______
- Consider the following three statements for the function $f : (0, \infty) \rightarrow \mathbb{R}$ defined by $f(x) = |\log_e x| - |x - 1|$:
(I) $f$ is differentiable at all $x > 0$.
(II) $f$ is increasing in $(0, 1)$.
(III) $f$ is decreasing in $(1, \infty)$.
Then. - If the domain of the function $f(x) = \sin^{-1} \left( \frac{1}{x^2 - 2x - 2} \right)$ is $(-\infty, \alpha] \cup [\beta, \gamma] \cup [\delta, \infty)$, then $\alpha + \beta + \gamma + \delta$ is equal to
Let $[t]$ denote the greatest integer less than or equal to $t$. If the function
$f(x) = \begin{cases} b^2 \sin \left( \frac{\pi}{2} \left[ \frac{\pi}{2} (\cos x + \sin x) \cos x \right] \right), & x < 0 \\ \frac{\sin x - \frac{1}{2} \sin 2x}{x^3}, & x > 0 \\ a, & x = 0 \end{cases}$
is continuous at $x = 0$, then $a^2 + b^2$ is equal to
- Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable function such that the quadratic equation $f(x)m^2 - 2f'(x)m + f''(x) = 0$ in m, has two equal roots for every $x \in \mathbb{R}$. If $f(0) = 1, f'(0) = 2$, and $(\alpha, \beta)$ is the largest interval in which the function $f(\log_e x - x)$ is increasing, then $\alpha + \beta$ is equal to ____________.
