Sure! Let's find the simplified form of the given function:
[tex]$
f(x)=\frac{1}{\left(\frac{3}{2}\right)^{2-x}}
$[/tex]
Step 1: Understand the function notation.
The given function can be interpreted as:
[tex]$
f(x) = \frac{1}{\left(\left(\frac{3}{2}\right)\right)^{2-x}}
$[/tex]
Step 2: Recall the rule of exponents that says [tex]\(\frac{a}{b} = a \cdot b^{-1}\)[/tex].
```math
\frac{3}{2} = 1.5 (approximated value)
```
Therefore:
[tex]$
\left(\frac{3}{2}\right)^{2-x} = 1.5^{2-x}
$[/tex]
Step 3: Apply the law of exponents:
[tex]$
f(x)=\frac{1}{1.5^{2-x}}
```
which can be written as:
```math
f(x)=1.5^{- (2-x)}
```
Simplifying further gives:
$[/tex]f(x)=1.5^{x-2}
[tex]$
Therefore, the simplified form of the given function is:
$[/tex]
\boxed{1.5^{x-2}}
[tex]$[/tex]
