Question

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

• A. True
• B. False

Answer 1

The correct option is A True

$Iff\left(a\right)=g\left(a\right)$ for all $aϵA$ then $f$ and $g$ are equal.

Given: $A=\{-1,0,1,2\}$ and $B=\{-4,-2,0,2\}$

$f\left(x\right)=x^2-x,xϵA$ and $g\left(x\right)=2|x-\frac{1}{2}|-1,xϵA$

Check$x=-1$

$f(-1)={(-1)}^2-(-1)=2g(-1)=2|x-\frac{1}{2}|-1=2\left(\frac{3}{2}\right)-1=2\Rightarrow f(-1)=g(-1)$

Check$x=0$

$f\left(0\right)=0^2-0=0g\left(0\right)=2|0-\frac{1}{2}|-1=2\left(\frac{1}{2}\right)-1=0f\left(0\right)=g\left(0\right)$

Check$x=1$

$f\left(1\right)=1^2-1=0g\left(1\right)=2|1-\frac{1}{2}|-1=1-1=0f\left(1\right)=g\left(1\right)$

Check$x=2$

$f\left(2\right)=2^2-2=4-2=2g\left(2\right)=2|2-\frac{1}{2}|-1=2\left(\frac{3}{2}\right)-1=2f\left(2\right)=g\left(2\right)$

Since $f\left(a\right)=g\left(a\right)\forall aϵA$, $f$ and $g$ are equal.

$\therefore$ The statement is true.