Question

Find the exponential function f(x)=ab^x that satisfies the conditions f(0)=-4 and f(-2)=-64.

f(x)=

Answer 1

`f(x) = ab^x~\hfill \begin{array}{llll}\textit{we know that}~f(0)=-4\begin{cases}x=0\\f(x)=-4\end{cases}\\\\-4=ab^0\implies -4=a(1)\implies -4=a\end{array}\\\\\\\stackrel{therefore}{f(x)=-4b^x}~\hfill \textit{we also know that}~f(-2)=-64\begin{cases}x=-2\\f(x)=-64\end{cases}`

`-64=-4b^(-2)\implies \cfrac{-64}{-4}=b^(-2)\implies 16=b^(-2)\implies 16=\cfrac{1}{b^2} \\\\\\ b^2=\cfrac{1}{16}\implies b=\sqrt{\cfrac{1}{16}}\implies b = \cfrac{√(1)}{√(16)}\implies b = \cfrac{1}{4} \\\\[-0.35em] ~\dotfill\\\\ ~\hfill f(x)=-4\left( (1)/(4) \right)^x~\hfill`