To determine: C and a so that f(x)=Ca^{x} satisfies given conditions.

Tazmin Horton 2021-09-24 Answered
To determine: C and a so that \(\displaystyle{f{{\left({x}\right)}}}={C}{a}^{{{x}}}\) satisfies given conditions.
Given: \(\displaystyle{f{{\left({0}\right)}}}={3};{f{{\left({3}\right)}}}={24}\)

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Expert Answer

StrycharzT
Answered 2021-09-25 Author has 12588 answers
Formula used:
If \(\displaystyle{b}^{{{x}}}={a}^{{{x}}}\) with \(\displaystyle{a},{b}{>}{0}\) then \(\displaystyle{b}={a}\).
Calculation:
Consider the given function \(\displaystyle{f{{\left({x}\right)}}}={C}{a}^{{{x}}}\). Substituting \(\displaystyle{x}={0}\) we get
This gives, \(\displaystyle{C}={3}\).
Substituting \(\displaystyle{x}={3}\in{f{{\left({x}\right)}}}={3}{a}^{{{x}}}\) we get
\(\displaystyle{f{{\left({3}\right)}}}={24}={3}{a}^{{{3}}}\)
\(\displaystyle{a}^{{{3}}}={\frac{{{24}}}{{{3}}}}\)
\(\displaystyle{a}^{{{3}}}={8}={2}^{{{3}}}\)
This gives \(\displaystyle{a}={3}\).
Conclusion:
From the given conditions we find \(\displaystyle{f{{\left({x}\right)}}}={3}{\left({2}^{{{x}}}\right)}\)
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