Answer
$f(x)=10\displaystyle \sqrt{2}(\frac{\sqrt{2}}{2})^{x}$
Work Step by Step
Exponential models have form: $f(x)=y=Ab^{x}$
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Given
$f(1)=10\Rightarrow\quad 10=Ab^{1},$
Given
$f(3)=5\Rightarrow\quad 5=Ab^{3},$
Dividing the second equation with the first,
$\displaystyle \frac{5}{10}=\frac{Ab^{3}}{Ab}$
$\displaystyle \frac{1}{2}=b^{2}\Rightarrow b=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$
Back-substitute: $10=A(\displaystyle \frac{1}{\sqrt{2}})\Rightarrow A=10\sqrt{2}$
Thus,
$f(x)=10\displaystyle \sqrt{2}(\frac{\sqrt{2}}{2})^{x}$
