Question

Let $f=\left\{\right(1,1),(2,3),(0,-1),(-1,3\left)\right\}$ be a linear function from $z$ to $z$ , find $f\left(x\right)$ where $x$ is the set of integers.

Answer 1

Since $f$ is a linear function.

Let $f\left(x\right)=mx+c$

	$x$	$y$	$y=mx+c$
$(1,1)$	$1$	$1$	$1=m\times 1+c\Rightarrow 1=m+c$
$(2,3)$	$2$	$3$	$3=m\times 2+c\Rightarrow 3=2m+c$

Solving $m+c=1$ .....$\left(1\right)$

$2m+c=3$ .......$\left(2\right)$

Equation $\left(2\right)-\left(1\right)$

$\Rightarrow 2m+c-(m+c)=3-1$

$\Rightarrow 2m+c-m-c=2$

$\Rightarrow m=2$

Put $m=2$ in $\left(1\right)$

$m+c=1$

$\Rightarrow 2+c=1$

$\Rightarrow c=1-2=-1$

So,$f\left(x\right)=mx+c=2x-1$