Question

Let A={-1,0,1,2}, B={-4,-2, 0,2} and $f,g:A\to B$ be the function defined by $f\left(x\right)=x^2-x,x\in A$ and $g\left(x\right)=2|x-\frac{1}{2}|-1,x\in A.$
Are f and g equal ? Justify your answer.

Answer 1

Given that $f,g:A\to B$ are defined by $f\left(x\right)=x^2-x,x\in A$ and $g\left(x\right)=2|x-\frac{1}{2}|-1,x\in A.$
It is observed that $f(-1)=(-1{)}^2-(-1)=1+1=2$
and $g(-1)=2\left|\right(-1)-\frac{1}{2}|-1=2\left(\frac{3}{2}\right)-1=3-1=2\Rightarrow f(-1)=g(-1)$
Next $f\left(0\right)=(0{)}^2-0=0andg(0)=2|0-\frac{1}{2}|-1=2(\frac{1}{2})-1=1-1=0$
$\Rightarrow f\left(0\right)=g\left(0\right)$
Next, $f\left(1\right)=(1{)}^2-1=1-1=0$
and $g\left(1\right)=2|1-\frac{1}{2}|-1=2\left(\frac{1}{2}\right)-1=1-1=0$ $\Rightarrow f\left(1\right)=g\left(1\right)$
Next, $f\left(2\right)=(2{)}^2-2=4-2=2$
and $g\left(2\right)=2|2-\frac{1}{2}|-1=2\left(\frac{3}{2}\right)-1=3-1=2$ $\Rightarrow f\left(2\right)=g\left(2\right)$
$\therefore f\left(a\right)=g\left(a\right)\forall a\in A$. Hence, the function f and g are equal.