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Sagot :
Solution:
Using the exponential function below
[tex]f(x)=ab^x[/tex]Where
[tex]\begin{gathered} a\text{ is the initial amount} \\ b\text{ is the base function \lparen exponential growth or decay\rparen} \\ x\text{ is the power/exponent} \end{gathered}[/tex]For exponential decay,
[tex]b=1-r[/tex]For exponential growth
[tex]b=1+r[/tex]Suppose, we are given a function
[tex]f(x)=3(2^x)[/tex]The function above will have an inverse.
To find the inverse of the function, where f(x) = y,
Firstly, replace x with y and y with x as shown below
[tex]\begin{gathered} y=3(2^x) \\ Replace\text{ x with y and y with x} \\ x=3(2^y) \end{gathered}[/tex]Then, solve for y
[tex]\begin{gathered} x=3(2^y) \\ Divide\text{ both sides by 3} \\ \frac{x}{3}=\frac{3(2^y)}{3} \\ \frac{x}{3}=2^y \\ Applying\text{ logarithm to both sides} \\ \ln(\frac{x}{3})=\ln(2^y) \\ \ln(\frac{x}{3})=y\ln2 \\ Divide\text{ both sides by }\ln2 \\ \frac{\ln(\frac{x}{3})}{\ln2}=\frac{y\ln2}{\ln2} \\ y=\frac{\ln(\frac{x}{3})}{\ln2} \end{gathered}[/tex]Hence, the inverse is
[tex]y=\frac{\ln(\frac{x}{3})}{\ln2}[/tex]
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