If f be a greatest integer function and g be an absolute value function, find the value of

Solution:

To find: (fog) $\left(\frac{-3}{2}\right)+($ gof $)\left(\frac{4}{3}\right)$

Formula used: (i) f o g = f(g(x))

(ii) g o f = g(f(x))

Given: (i) f is a greatest integer function

(ii) g is an absolute value function

$f(x)=[x]$ (greatest integer function)

$g(x)=|x|$ (absolute value function)

$f\left(\frac{4}{3}\right)=\left[\frac{4}{3}\right]=1 \ldots$ (i)

$g\left(\frac{-3}{2}\right)=\left|\frac{-3}{2}\right|=1.5 \ldots$ (ii)

Now, for (fog) $\left(\frac{-3}{2}\right)+($ gof $)\left(\frac{4}{3}\right)$

$\Rightarrow f\left(g\left(\frac{-3}{2}\right)\right)+g\left(f\left(\frac{4}{3}\right)\right)$

Substituting values from (i) and (ii)

$\Rightarrow f(1.5)+g(1)$

$\Rightarrow[1.5]+|1|$

Ans) 2