Question

Find an exponential function in the news/media online. What is the function and what do the variables represent? Does it have an inverse? if so, what is the inverse?( this is all one question im just on my laptop and cant upload a picture) thank you

Answer 1

Solution:

Using the exponential function below

$f(x)=ab^x$

Where

$\begin{gathered} a\text{ is the initial amount} \\ b\text{ is the base function \lparen exponential growth or decay\rparen} \\ x\text{ is the power/exponent} \end{gathered}$

For exponential decay,

$b=1-r$

For exponential growth

$b=1+r$

Suppose, we are given a function

$f(x)=3(2^x)$

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

$\begin{gathered} y=3(2^x) \\ Replace\text{ x with y and y with x} \\ x=3(2^y) \end{gathered}$

Then, solve for y

$\begin{gathered} x=3(2^y) \\ Divide\text{ both sides by 3} \\ \frac{x}{3}=\frac{3(2^y)}{3} \\ \frac{x}{3}=2^y \\ Applying\text{ logarithm to both sides} \\ \ln(\frac{x}{3})=\ln(2^y) \\ \ln(\frac{x}{3})=y\ln2 \\ Divide\text{ both sides by }\ln2 \\ \frac{\ln(\frac{x}{3})}{\ln2}=\frac{y\ln2}{\ln2} \\ y=\frac{\ln(\frac{x}{3})}{\ln2} \end{gathered}$

Hence, the inverse is

$y=\frac{\ln(\frac{x}{3})}{\ln2}$