Let \( f : R \to R \) and \( g : R \to R \) be defined as . Then, \( gof : R \to R \) is:
neither one-one nor onto
both one-one and onto
onto but not one-one
one-one but not onto
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 __________.
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:
. Then, \( gof : R \to R \) is:
then range of \( (f og)(x) \) is: