Let \( n(A)=m \) and \( n(B)=n \). Then the total number of non-empty relations that can be defined from A to B is
\( m^n \)
\( n^m - 1 \)
\( mn - 1 \)
\( 2^{mn} - 1 \)
If \( [x]^2 - 5[x] + 6 = 0 \), where \([.]\) denotes the greatest integer function, then
\( x \in [3,4] \)
\( x \in (2,3] \)
\( x \in [2,3] \)
\( x \in [2,4] \)
The range of \( f(x)=\dfrac{1}{1-2\cos x} \) is
\( \left[\dfrac{1}{3},1\right] \)
\( [-1,\tfrac{1}{3}] \)
\( (-\infty,-1] \cup [\tfrac{1}{3},\infty) \)
\( [-\tfrac{1}{3},1] \)
Let \( f(x)=\sqrt{1+x^2} \). Then
\( f(xy)=f(x)f(y) \)
\( f(xy)\ge f(x)f(y) \)
\( f(xy)\le f(x)f(y) \)
None of these
The domain of \( \sqrt{a^2-x^2} \) where \( a>0 \) is
\( (-a,a) \)
\( [-a,a] \)
\( [0,a] \)
\( (-a,0] \)
If \( f(x)=ax+b \), where \( a,b \) are integers, \( f(-1)=-5 \) and \( f(3)=3 \), then \( a \) and \( b \) are
\( a=-3, b=-1 \)
\( a=2, b=-3 \)
\( a=0, b=2 \)
\( a=2, b=3 \)
The domain of the function defined by \( f(x)=\sqrt{4-x}+\dfrac{1}{\sqrt{x^2-1}} \) is
\( (-\infty,-1) \cup (1,4] \)
\( (-\infty,-1] \cup (1,4] \)
\( (-\infty,-1) \cup [1,4] \)
None of these
The domain and range of the real function \( f(x)=\dfrac{4-x}{x-4} \) is
Domain = \( \mathbb{R} \), Range = \{ -1, 1 \}
Domain = \( \mathbb{R}-\{1\} \), Range = \( \mathbb{R} \)
Domain = \( \mathbb{R}-\{4\} \), Range = \{ -1 \}
Domain = \( \mathbb{R}-\{4\} \), Range = \{ -1,1 \}
The domain and range of the function \( f(x)=\sqrt{x-1} \) is
Domain = \( (1,\infty) \), Range = \( (0,\infty) \)
Domain = \( [1,\infty) \), Range = \( 0,\infty) \)
Domain = \( [1,\infty) \), Range = \( [0,\infty) \)
Domain = \( [1,\infty) \), Range = \( [0,\infty) \)
The domain of the function \( f(x)=\dfrac{x^2+2x+1}{x^2-x-6} \) is
\( \mathbb{R}-\{3,-2\} \)
\( \mathbb{R}-\{-3,2\} \)
\( \mathbb{R}-[3,-2] \)
\( \mathbb{R}-(3,-2) \)
The domain and range of the function \( f(x)=2-|x-5| \) is
Domain = \( \mathbb{R}^+ \), Range = \((-\infty,1] \)
Domain = \( \mathbb{R} \), Range = \((-\infty,2] \)
Domain = \( \mathbb{R} \), Range = \((-\infty,2) \)
Domain = \( \mathbb{R}^+ \), Range = \((-\infty,2] \)
The domain for which the functions \( f(x)=3x^2-1 \) and \( g(x)=3+x \) are equal is
\( \{-1,\dfrac{4}{3}\} \)
\( \left\{-1,\dfrac{4}{3}\right\} \)
\( \{-1,\dfrac{4}{3}\} \)
\( \{-1,\dfrac{4}{3}\} \)