We want to find the inverse of the function
f of x equals three times e raised to the power of two x.
The first thing we'll do is determine
if the given function has an inverse function.
Only one-to-one functions have inverse functions.
A one-to-one function is a function
where every output or every function value
has only one input or one x-value.
So every y-value can only have one corresponding x-value.
One way to determine whether a function is one-to-one
is to use what's called a horizontal line test.
To perform the horizontal line test,
we begin by graphing the given function,
as we see here in blue.
And if a horizontal line never intersects
the graph in more than one point,
then the function passes the horizontal line test
and the function is one-to-one.
Looking at our function here,
notice how no horizontal line would ever
intersect this graph in more than one point.
And therefore this function passes the horizontal line test
and it's one-to-one, which means
it does have an inverse function.
So now that we know it has an inverse function,
we can find it.
Let's begin by replacing f of x with y.
So we can write the given function
as y equals three times e raised to the power of two x.
And now to find the inverse function,
we interchange the x and y variables.
And therefore the inverse function
would be x equals three
times e raised to the power of two y.
But now we need to solve this for y.
Let's begin by dividing both sides by three.
So now we have x divided by three
equals e raised to the power of two y.
Notice how the variable y's in the exponent.
So if we take the natural log of both sides of the equation
here and here,
we can apply the power property of logarithms,
meaning on the right side, we can take the exponent
and write it as a product with natural log e.
So we'd have natural log of x divided by three equals
two y times natural log e.
We didn't have to use natural log e
in order to apply this log property.
But notice how because this is base e,
natural log e is equal to one,
so this simplifies nicely to just natural log
x divided by three equals two y.
So now to solve for y, we would divide both sides by two.
So we have y is equal to this quotient,
where y is the inverse function.
So we'll use the inverse function notation for y
and we'll say f inverse of x equals-
We could leave this as the quotient,
but dividing by two is equivalent to multiplying by 1/2.
So let's write this as 1/2 times natural log
of x divided by three.
This would be our inverse function.
But I do want to mention one more thing.
Notice how we have natural log of a quotient.
So we could apply another log property
and write this as a difference of two logs.
This is equivalent to f inverse of x equals,
and we'd have 1/2 times
natural log x minus natural log three.
And of course we could also distribute the 1/2.
But to match our homework,
I'll go ahead and leave it in this form here.
And now before we go, to check our answer,
let's go ahead and graph our function
and the inverse function on the same coordinate plane.
And we'll also graph the line y equals x.
So here's the given function in blue.
Here's the graph of the inverse function in red.
The reason we also want to graph y equals x
is because remember, a function and its inverse
are symmetrical across the line y equals x.
And notice how if we were to fold this
across the line y equals x,
the two functions would match up perfectly,
verifying we found the inverse function correctly.
I hope you found this helpful.