Exercise 1

Let $Y=\left\{n^2:n\in N\right\}\subset N$. Consider $f:N\to Y$ as $f\left(n\right)=n^2$. Show that $f$ is invertible. Find the inverse of $f$.

Let $f:N\mathit{\to }Y$ be a function defined as $f\left(x\right)=4x+3$, where, $Y=\{y\in N:y=4x+3$ for some $x\in N$}. Show that $f$ is invertible. Find the inverse.

If $y=\frac{x^2+2x+1}{x^2+2x+7}$, then inverse function $x$ is defined only when

Let $f\left(x\right)$ and $g\left(x\right)$ be two real polynomials of degree $2$ and $1$ respectively. If $f\left(g\left(x\right)\right)=8x^2-2x,$ and $g\left(f\left(x\right)\right)=4x^2+6x+1,$ then find the value of $f\left(2\right)+g\left(2\right).$ 

Consider functions $f$ and $g$ such that composite $g$ of is defined and is one-one. Are $f$ and $g$ both necessarily one-one.

The relation R in the set $\left\{1,2,3\right\}$ given by $R=\left\{(1,3),(3,1)\right\}$ then R is 

Define the relation $R$ in the set $N\times N$ as follows:

For $\left(a,b\right),\left(c,d\right)\in N\times N,\left(a,b\right)R\left(c,d\right)$ iff $ad=bc.$ Prove that $R$ is an equivalence relation in $N\times N.$

Let $N$ denote the set of all natural number and $R$ is a relation on $N\times N$. Which of the following is an equivalence relation $?$