To determine which expression is an equivalent representation of the function [tex]\( f(x) = \frac{3}{2} \left( \frac{5}{2} \right)^{x-1} \)[/tex], let's examine and verify each given option step by step.
### Option 1: [tex]\( f(x+1) = \frac{5}{2} f(x) \)[/tex]
First, calculate [tex]\( f(x+1) \)[/tex]:
[tex]\[ f(x+1) = \frac{3}{2} \left( \frac{5}{2} \right)^{(x+1)-1} = \frac{3}{2} \left( \frac{5}{2} \right)^x \][/tex]
Now, calculate [tex]\( \frac{5}{2} f(x) \)[/tex]:
[tex]\[ \frac{5}{2} f(x) = \frac{5}{2} \times \frac{3}{2} \left( \frac{5}{2} \right)^{x-1} = \left( \frac{5}{2} \times \frac{3}{2} \right) \left( \frac{5}{2} \right)^{x-1} = \frac{15}{4} \left( \frac{5}{2} \right)^{x-1} \][/tex]
Simplify the right-hand side:
[tex]\[ \frac{5}{2} f(x) = \frac{3}{2} \left( \frac{5}{2} \right) \left( \frac{5}{2} \right)^{x-1} = \frac{3}{2} \left( \frac{5}{2} \right)^x \][/tex]
Since [tex]\( f(x+1) = \frac{3}{2} \left( \frac{5}{2} \right)^x \)[/tex], it is clear that:
[tex]\[ f(x+1) = \frac{5}{2} f(x) \][/tex]
Therefore, Option 1 is correct.
### Option 2: [tex]\( f(x) = \frac{5}{2} f(x+1) \)[/tex]
Let's rearrange Option 1 for Option 2:
[tex]\[ f(x+1) = \frac{5}{2} f(x) \][/tex]
To isolate [tex]\( f(x) \)[/tex], multiply both sides by [tex]\( \frac{2}{5} \)[/tex]:
[tex]\[ f(x) = \frac{2}{5} f(x+1) \][/tex]
Option 2 states [tex]\( f(x) = \frac{5}{2} f(x+1) \)[/tex], which is not consistent with the rearrangement. Thus, Option 2 is incorrect.
### Option 3: [tex]\( f(x+1) = \frac{3}{2} f(x) \)[/tex]
Calculate [tex]\( \frac{3}{2} f(x) \)[/tex]:
[tex]\[ \frac{3}{2} f(x) = \frac{3}{2} \times \frac{3}{2} \left( \frac{5}{2} \right)^{x-1} = \left( \frac{3}{2} \times \frac{3}{2} \right) \left( \frac{5}{2} \right)^{x-1} = \frac{9}{4} \left( \frac{5}{2} \right)^{x-1} \][/tex]
Since [tex]\( f(x+1) = \frac{3}{2} \left( \frac{5}{2} \right)^x \)[/tex], it is clear that:
[tex]\[ f(x+1) \neq \frac{3}{2} f(x) \][/tex]
Hence, Option 3 is incorrect.
### Option 4: [tex]\( f(x) = \frac{3}{2} f(x+1) \)[/tex]
Let's rearrange Option 1 to test Option 4:
[tex]\[ f(x+1) = \frac{5}{2} f(x) \][/tex]
To isolate [tex]\( f(x) \)[/tex], multiply both sides by [tex]\( \frac{2}{5} \)[/tex]:
[tex]\[ f(x) = \frac{2}{5} f(x+1) \][/tex]
Option 4 states [tex]\( f(x) = \frac{3}{2} f(x+1) \)[/tex], which is clearly not the same. Therefore, Option 4 is also incorrect.
Conclusively, the only correct equivalent representation is:
[tex]\[ \boxed{f(x+1) = \frac{5}{2} f(x)} \][/tex]
