Standard functions

From the given condition:

f(x + y) = f(x) f(y)

This is an exponential-type function, so we assume:

f(x) = a^x

Now:

f(100x) = a^(100x)

But here’s the key step you missed:

f(x) = a^x

⇒ a = (f(x))^(1/x)

Now substitute back:

f(100x) = a^(100x)

= [(f(x))^(1/x)]^(100x)

= (f(x))^100

So your mistake was directly writing:

a^(100x) = (f(x))^(100x)

But:

a = (f(x))^(1/x), not f(x)

Final answer:

(f(x))^100