Question

Let $f=\left\{\right(1,1),(2,3),(0,-1),(-1,-3\left)\right\}$ be a linear function from $Z$ into $Z.$ Find $f\left(x\right)$

Answer 1

Step 1: Given
$f=\left\{\right(1,1),(2,3),(0,-1),(-1,-3\left)\right\}$

Step $2:$ Assume function
Let $f$ be a linear function, $f\left(x\right)=ax+b$

Step $3:$ Finding function
Given points satisfied the function $f\left(x\right)$.
Put $x=1$ and $y=1$ in $f\left(x\right)$, we get
$\Rightarrow 1=a+b\cdots \left(i\right)$
Put $x=0$ and $y=-1$ in $f\left(x\right)$, we get
$\Rightarrow -1=a\cdot 0+b\Rightarrow b=-1\cdots \left(ii\right)$
Put the value of $b$ in equation $\left(i\right),$ we get
$a-1=1\Rightarrow a=2$
So, $a=2,b=-1$ put in $f\left(x\right)$.
Hence, function $f\left(x\right)$ is equal to $2x-1$.