List of top Mathematics Questions on Relations and functions

If $f:\mathbb{R}-\left\{\frac{2a}{3}\right\}\rightarrow \mathbb{R}$ is a function defined by \[ f(x)=\frac{2x+a}{3x-2a} \] and $(f\circ f)(x)=x$ for all $x\in \mathbb{R}-\left\{\frac{2a}{3}\right\}$, then $f(3)=$}

If $f:\mathbb{R}\rightarrow\mathbb{R}$ and $g:\mathbb{R}\rightarrow\mathbb{R}$ are defined by \[ f(x)=|x+1| \] and \[ g(x)= \begin{cases} e^{-x}, & x\le 0\\ x-1, & x>0 \end{cases} \] then $(f\circ g)(-2)+(g\circ f)(2)=$}

The domain of the real valued function \[ f(x)=\cos^{-1}\!\left(\log_{5}\frac{x}{5}\right)+\log_{5}\!\left(\cos^{-1}\frac{x}{5}\right) \] is:

If \(f(x)\) is a linear polynomial such that \[ f(ax+by)=af(x)+bf(y) \] for all \(x,y\in\mathbb{R}\). If \(f(0)=7\) and \(f'(0)=5\), then \[ \frac{f(1)+f(-1)}{a+b}= \]

If \(f:\mathbb R\to\mathbb R\) is defined by \[ f(x)=|x| \] and \(A=(0,1)\), then \(f^{-1}(A)\) is:

If the function \(f:\left[0,\dfrac{\pi}{2}\right]\to R\) is given by \(f(x)=\cos x\) and the function \(g:\left[0,\dfrac{\pi}{2}\right]\to R\) is given by \(g(x)=\sin x\), then prove that \(f\) and \(g\) are one-one, but \(f+g\) is not one-one.

If the ordered pairs \((2x-3,-5)\) and \((x,y-1)\) are equal, then find the value of \(x\) and \(y\).

The function \(f: R \to R\) defined by \(f(x) = 3x + 5,\ \forall x \in R\) is:

A relation \(R\) is defined in \(N\times N\) as follows: \((a,b)\,R\,(c,d)\) if and only if \(ad=bc\). Prove that \(R\) is an equivalence relation.

\(f:X \to Y\) will be an onto function if and only if its range is

If functions \(f:R\to R\) and \(g:R\to R\) are defined respectively as \(f(x)=\cos x\) and \(g(x)=3x^2\), then find \(gof\) and \(fog\).

The relation \(R=\{(a,b): a\le b^2\}\) is defined in the set of real numbers. Then \(R\) is:

If the function \(f:N\to N\) is defined as \(f(x)=x^2\), then \(f\) is:

If \(f:\mathbb{R}\to\mathbb{R}\) and \(g:\mathbb{R}\to\mathbb{R}\) be functions such that \(f(x)=\cos x\) and \(g(x)=3x^3\), then find \(f\circ g\).

If \(X=\{1,2,3\}\) and \(Y=\{a,b\}\), then find the number of functions from \(X\) to \(Y\).

The function \(f:Z\to Z\) defined by \(f(x)=x^3\ \forall x\in Z\) is:

If \(A=\{1,2,3\}\), then the number of equivalence relations on \(A\) containing \((1,2)\) is:

Prove that the relation given by \(R=\{(1,1),(2,2),(3,3),(1,2),(2,3)\}\) in the set \(\{1,2,3\}\) is reflexive but neither symmetric nor transitive.

A relation \(R\) is defined in the set \(N\) as \(R=\{(x,y):y=x+5,\,y>5\}\). Then which of the following is correct?

If \(f:R\to R\) is given by \(f(x)=(5-x^{5})^{1/5}\), then \(f\circ f(x)\) is equal to:

If \(f(x)=\sin x\), \(x\in[0,\pi/2]\) and \(g(x)=\cos x\), \(x\in[0,\pi/2]\), prove that \((f+g)\) is not one-one.

Prove that the function \(f:R\to\{x\in R:-1<x<1\}\), where \(f(x)=\dfrac{2x}{1+|x|}\), \(x\in R\), is one-one.

If \(f:R\to R\) and \(g:R\to R\) are defined by \(f(x)=\cos x\) and \(g(x)=3x^2\) respectively, find \(gof\).

The function \(f(x)=2x\), \(x\in R\) is: