Question

Let the expression $\left[ x \right]$ denote the greatest integer $\le x$ . If $f\left( x \right)=\left[ x \right]$ and $g\left( x \right)=\left| x \right|$ , then the value of $f\left( g\left( \dfrac{8}{5} \right) \right)-g\left( f\left( -\dfrac{8}{5} \right) \right)$ is,
A. $2$
B. $-2$
C. $1$
D. $0$
E. $-1$

Answer 1

We evaluate the expression term by term. We know that $f\left( g\left( x \right) \right)$ can be evaluate by evaluating $g\left( x \right)$ first and then the function f. $f\left( g\left( \dfrac{8}{5} \right) \right)$ gives $f\left( \left| \dfrac{8}{5} \right| \right)=\left[ \dfrac{8}{5} \right]=1$ . Similarly, $g\left( f\left( -\dfrac{8}{5} \right) \right)$ gives $g\left( \left[ -\dfrac{8}{5} \right] \right)=g\left( -2 \right)=\left| -2 \right|=2$ . Subtracting $2$ from $1$ , we get the answer.

Complete step by step solution:
The given expression that we are given to evaluate is,
$f\left( g\left( \dfrac{8}{5} \right) \right)-g\left( f\left( -\dfrac{8}{5} \right) \right)$
The above expression contains a function of function expression. Now, we know that in order to evaluate such complex function types, we first evaluate the innermost function and then proceed to the outermost function.
Let us first evaluate the first term which is $f\left( g\left( \dfrac{8}{5} \right) \right)$ . This term can be rewritten as,
$\Rightarrow f\left( \left| \dfrac{8}{5} \right| \right)$
Now the absolute value of $\dfrac{8}{5}$ is $\dfrac{8}{5}$ itself. So, the term becomes,
$\Rightarrow f\left( \dfrac{8}{5} \right)=\left[ \dfrac{8}{5} \right]$
Now, $\dfrac{8}{5}$ can be written as $1.6$ . So, the term becomes,
$\Rightarrow \left[ 1.6 \right]=1$
Let us now evaluate the second term which is $g\left( f\left( -\dfrac{8}{5} \right) \right)$ . This term can be rewritten as,
$\Rightarrow g\left( \left[ -\dfrac{8}{5} \right] \right)$
Now, $\dfrac{8}{5}$ can be written as $1.6$ . So, the term becomes,
$\Rightarrow g\left( \left[ -1.6 \right] \right)$
The least integer which is less than or equal to $-1.6$ is $-2$ . So, the term becomes,
$\Rightarrow g\left( -2 \right)=\left| -2 \right|$
Now the absolute value of $-2$ is $2$ itself. So, the term becomes,
$\Rightarrow 2$
The entire expression becomes,
$\Rightarrow f\left( g\left( \dfrac{8}{5} \right) \right)-g\left( f\left( -\dfrac{8}{5} \right) \right)=1-2=-1$
Thus, we can conclude that the value of $f\left( g\left( \dfrac{8}{5} \right) \right)-g\left( f\left( -\dfrac{8}{5} \right) \right)$ is $-1$ which is option E.

Note: Function of function problems are very easy once we get to know how to solve them. The real confusion comes from the box $\left[ {} \right]$ function for negative numbers. We should remember to take the integer which just lies on the left of the number on the number line for negative numbers.