Answer :
We start with the function
[tex]$$
f(x)=3\cdot2^{(3-x)}.
$$[/tex]
Our first step is to rewrite the expression using properties of exponents. Notice that
[tex]$$
2^{(3-x)} = 2^3 \cdot 2^{-x}.
$$[/tex]
So, we have
[tex]$$
f(x) = 3\cdot2^3\cdot2^{-x}.
$$[/tex]
Since [tex]$2^3=8$[/tex], multiplying [tex]$3$[/tex] by [tex]$8$[/tex] gives
[tex]$$
3\cdot8=24.
$$[/tex]
Thus, the function becomes
[tex]$$
f(x)=24\cdot2^{-x}.
$$[/tex]
Recall that [tex]$2^{-x}$[/tex] can be written as [tex]$\left(\frac{1}{2}\right)^x$[/tex]. Therefore, we obtain
[tex]$$
f(x)=24\cdot\left(\frac{1}{2}\right)^x.
$$[/tex]
Thus, the equivalent expression for the function [tex]$f(x)$[/tex] is
[tex]$$
24 \cdot \left(\frac{1}{2}\right)^x.
$$[/tex]
This corresponds to option (b).
[tex]$$
f(x)=3\cdot2^{(3-x)}.
$$[/tex]
Our first step is to rewrite the expression using properties of exponents. Notice that
[tex]$$
2^{(3-x)} = 2^3 \cdot 2^{-x}.
$$[/tex]
So, we have
[tex]$$
f(x) = 3\cdot2^3\cdot2^{-x}.
$$[/tex]
Since [tex]$2^3=8$[/tex], multiplying [tex]$3$[/tex] by [tex]$8$[/tex] gives
[tex]$$
3\cdot8=24.
$$[/tex]
Thus, the function becomes
[tex]$$
f(x)=24\cdot2^{-x}.
$$[/tex]
Recall that [tex]$2^{-x}$[/tex] can be written as [tex]$\left(\frac{1}{2}\right)^x$[/tex]. Therefore, we obtain
[tex]$$
f(x)=24\cdot\left(\frac{1}{2}\right)^x.
$$[/tex]
Thus, the equivalent expression for the function [tex]$f(x)$[/tex] is
[tex]$$
24 \cdot \left(\frac{1}{2}\right)^x.
$$[/tex]
This corresponds to option (b).
