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If \( f(x)=\frac{4x+3}{6x-4}, \, x\neq \frac{2}{3} \) and \( (fof)(x)=g(x) \), where \( g:\mathbb{R}-\left\{\frac{2}{3}\right\}\to \mathbb{R}-\left\{\frac{2}{3}\right\} \), \( (gogog)(4) \) is equal to:

Let \( f:\mathbb{R}-\left\{\displaystyle\,\frac{-1}{2}\,\right\}\to \mathbb{R} \) and \( g:\mathbb{R}-\left\{\displaystyle\,\frac{-5}{2}\,\right\}\to \mathbb{R} \) be defined as \(\displaystyle f(x)=\frac{2x+3}{2x+1} \) and \(\displaystyle g(x)=\frac{|x|+1}{2x+5} \). Then, the domain of the function fog is:

Let \( f(x) = x^5 + 2x^3 + 3x + 1, \ x \in \mathbb{R} \), and \( g(x) \) be a function such that \( g(f(x)) = x \) for all \( x \in \mathbb{R} \). Then \( \dfrac{g(7)}{g'(7)} \) is equal to:

If the domain of the function \(\displaystyle f(x) = \frac{\sqrt{x^2 - 25}}{(4 - x^2)} + \log_{10}\!\left( x^2 + 2x - 15 \right) \) is \( (-\infty, \alpha) \cup (\beta, \infty) \), then \( \alpha^2 + \beta^3 \) is equal to:

If the domain of the function \( f(x)=\log_{e}\!\left(\dfrac{2x+3}{4x^{2}+x-3}\right)+\cos^{-1}\!\left(\dfrac{2x-1}{x+2}\right) \) is \( (\alpha,\beta] \), then the value of \( 5\beta-4\alpha \) is equal to:

If \( S = \{ a \in \mathbb{R} \ :|2a - 1| = 3\ [a] + 2\{a\} \} \), where [t] denotes the greatest integer less than or equal to \( t \) and \( \{t\} \) represents the fractional part of \( t \), then \( 72 \sum\limits_{a \in S} a \) is equal to \( \_\_\_\_\_\_\_\_\_\_\_ \).

Consider the function \( f:\mathbb{R}\to\mathbb{R} \) defined by \( f(x)=\dfrac{2x}{\sqrt{1+9x^{2}}} \). If the composition of \( (\underbrace{f \circ f \circ f \circ \dots \circ f}_{10\text{ times}})(x) = \frac{2^{10}x}{\sqrt{1 + 9\alpha x^2}} \), then the value of \( \sqrt{3\alpha+1} \) is equal to __________.

Let \( f : R \to R \) and \( g : R \to R \) be defined as . Then, \( gof : R \to R \) is:

fg

The number of distinct real roots of the equation \( |x|\,|x+2| - 5|x+1| - 1 = 0 \) is \( \_\_\_\_\_\_\_\_\_\_\_ \).

Let \( A = \{(x,y): 2x + 3y = 23, \, x,y \in \mathbb{N}\} \) and \( B = \{ x : (x,y) \in A \} \). Then the number of one-one functions from \( A \) to \( B \) is equal to ________.

If the function \( f:(-\infty,-1)\to (a,b] \) defined by \( f(x) = e^{x^3-3x+1} \) is one – one and onto, then the distance of the point \( P(2b+4,a+2) \) from the line \( x+e^{-3}y=4 \) is:

Let \( f , g : \mathbb{R} \to \mathbb{R} \) be defined as:

Then the function \( f(g(x)) \) is

Let \( f(x) = \frac{1}{7 - \sin 5x} \) be a function defined on \( \mathbb{R} \). Then the range of the function \( f(x) \) is equal to:

The function \( f(x)=\frac{x^{2}+2x-15}{x^{2}-4x+9},\ x\in \mathbb{R} \) is:

The function \( f: N-\{1\}\to N \); defined by \( f(n)= \) the highest prime factor of \( n \), is:

If the domain of the function \( \sin^{-1}\!\left(\dfrac{3x-22}{2x-19}\right) + \log_e\!\left(\dfrac{3x^2-8x+5}{x^2-3x-10}\right) \) is \( (\alpha,\ \beta] \), then \( 3\alpha + 10\beta \) is equal to:

Let \( f(x)=3\sqrt{x-2}+\sqrt{4-x} \) be a real valued function. If \( \alpha \) and \( \beta \) are respectively the minimum and the maximum values of \( f \), then \( \alpha^2+2\beta^2 \) is equal to:

Let \( A = \{1, 2, 3, \ldots, 7\} \) and let \( P(A) \) denote the power set of \( A \). If the number of functions \( f : A \to P(A) \) such that \( a \in f(a),\ \forall\, a \in A \) is \( m^n \), \( m \) and \( n \in \mathbb{N} \) and \( m \) is least, then \( m + n \) is equal to ______.

If a function \( f \) satisfies \( f(m+n)=f(m)+f(n) \) for all \( m,n\in \mathbb{N} \) and \( f(1)=1 \), then the largest natural number \( \lambda \) such that \( \displaystyle \sum_{k=1}^{2022} f(\lambda+k)\le (2022)^{2} \) is equal to ________.

Let \( [t] \) be the greatest integer less than or equal to \( t \). Let \( A \) be the set of all prime factors of \( 2310 \) and \( f : A \to \mathbb{Z} \) be the function \( f(x) = \left[ \log_{2} \left( x^{3} + \left[ \frac{x^{3}}{5} \right] \right) \right] \). The number of one-to-one functions from \( A \) to the range of \( f \) is:

then range of \( (f og)(x) \) is:

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JEE Main 2024 4th April Evening Shift

Let \( f(x)=3\sqrt{x-2}+\sqrt{4-x} \) be a real valued function. If \( \alpha \) and \( \beta \) are respectively the minimum and the maximum values of \( f \), then \( \alpha^2+2\beta^2 \) is equal to: