Question

Let $A = \{ 0,1,2 \}$, $B =\{ 4,2,0 \}$ and $ f,g $ : $ A \rightarrow B $ be the functions defined by $ f(x) = x^2-x $ and $ g(x) = 2|x-\frac{1}{2}|-1 $ Then,

• A. $ f = g $
• B. $ f = 2g $
• C. $ g = 2f $
• D. None of these

Answer 1

The Correct Option is A

We have $A=\left\{1, 0, 1, 2\right\}$
$B=\left\{4, 2, 0, 2\right\}$
and $f, g : A \rightarrow B$
Now, $f \left(x\right)=x^{2}-x$
$\Rightarrow f \left(0\right)=0-0=0$
and $f \left(1\right)=1-1=0\ldots\left(i\right) $
and $f \left(2\right)=2^{2}-2=2$
Also, $g\left(x\right)=2\left|x-\frac{1}{2}\right|-1$
$\Rightarrow g \left(0\right)=2\left|\frac{-1}{2}\right|-1=1-1=0$
and $ g\left(1\right)=2\left|1-\frac{1}{2}\right|-1$
$=1-=0 \ldots\left(ii\right)$
and $g \left(2\right)=2\left|2-\frac{1}{2}\right|-1$
$=\frac{2.3}{2}-1=2$
Hence from Eqs. $\left(i\right)$ and $\left(ii\right)$, we can say that
$f\left(x\right)=g\left(x\right)$
$\Rightarrow f=g$