[tex]f[/tex] has to be a power function in order to satisfy the recurrence pattern [tex]f(x + h) = f(x) \cdot f(h)[/tex]
In this case, we must assume a given function and check such assumption fulfill the given recurrence. Let suppose that [tex]f(x) = a^{x}[/tex], by algebra we have the following property:
[tex]f(x + h) = a^{x+h} = a^{x}\cdot a^{h}[/tex] (1)
And by the definition given in statement, we have the following conclusion:
[tex]f(x+h) = a^{x}\cdot a^{h} = f(x) \cdot f(h)[/tex] (2)
Therefore, [tex]f[/tex] has to be a power function in order to satisfy the recurrence pattern [tex]f(x + h) = f(x) \cdot f(h)[/tex]. [tex]\blacksquare[/tex]
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