If f be a greatest integer function and g be an absolute value function, find the value of
$(f \circ g)\left(\frac{-3}{2}\right)+(g \circ f)\left(\frac{4}{3}\right)$
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
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