To find an equivalent function to [tex]\(f(x) = 2(5)^{2x}\)[/tex], let's carefully analyze and rewrite the given function.
First, recall the properties of exponents. We have:
[tex]\[ f(x) = 2 \cdot (5)^{2x} \][/tex]
We can rewrite the exponent in a different form by using exponentiation properties. Specifically, [tex]\(a^{b \cdot c} = (a^b)^c\)[/tex]. Here, we can set [tex]\(a = 5\)[/tex] and [tex]\(b = 2x\)[/tex]:
[tex]\[ (5)^{2x} = [(5^2)]^x \][/tex]
Next, we calculate [tex]\(5^2\)[/tex]:
[tex]\[ 5^2 = 25 \][/tex]
Now substitute [tex]\(25\)[/tex] back into the function:
[tex]\[ f(x) = 2 \cdot (25)^x \][/tex]
So, the function [tex]\( f(x) = 2(5)^{2x} \)[/tex] is equivalent to [tex]\( f(x) = 2(25)^x \)[/tex].
From the given options:
1. [tex]\(f(x) = 50^x\)[/tex]
2. [tex]\(f(x) = 100^x\)[/tex]
3. [tex]\(f(x) = 2(25)^x\)[/tex]
4. [tex]\(f(x) = 4(25)^x\)[/tex]
The equivalent function to [tex]\( f(x) = 2(5)^{2x} \)[/tex] is:
[tex]\[ f(x) = 2(25)^x \][/tex]
Thus, the correct option is:
[tex]\[ \boxed{3} \][/tex]
