Model the following data using an exponential function of the form f(x)=Ab^xZ

sagnuhh 2021-09-29 Answered
Model the following data using an exponential function of the form \(\displaystyle{f{{\left({x}\right)}}}={A}{b}^{{x}}\) . Set up a system of equations, and solve it to get A and b.
f(x) is exponential and goes through the points (1, 2) and (4, 6).

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Expert Answer

toroztatG
Answered 2021-09-30 Author has 5688 answers
Given information:
The required model for the given data is exponential function of the form \(\displaystyle{f{{\left({x}\right)}}}={A}{b}^{{x}}\).
Find the function f(x) if it goes through (1,2) and (4,6):
The data points are (1,2) and (4,6).
The function f(x) is obtained as given below:
\(\displaystyle{f{{\left({x}\right)}}}={A}{b}^{{x}}\)
\(\displaystyle{2}={A}{b}^{{1}}\to\ {\left({1}\right)}\)
\(\displaystyle{A}{b}={2}\)
\(\displaystyle{6}={A}{b}^{{4}}\)
\(\displaystyle{A}{b}^{{4}}={6}\to\ {\left({2}\right)}\)
\(\displaystyle{\frac{{{\left({2}\right)}}}{{{\left({1}\right)}}}}\to{\frac{{{A}{b}^{{4}}}}{{{A}{b}}}}={\frac{{{6}}}{{{2}}}}={3}\)
\(\displaystyle{b}^{{3}}={3}\)
\(\displaystyle{b}={3}^{{{\frac{{{1}}}{{{3}}}}}}\)
\(\displaystyle{A}={\frac{{{2}}}{{{b}}}}={\frac{{{2}}}{{{3}^{{{\frac{{{1}}}{{{3}}}}}}}}}\)
\(\displaystyle{f{{\left({x}\right)}}}={\frac{{{2}}}{{{3}^{{{\frac{{{1}}}{{{3}}}}}}}}}{\left({3}^{{{\frac{{{1}}}{{{3}}}}}}\right)}^{{x}}\)
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