Here we're given F of X
equals 1/4 X cubed plus X plus one.
Part A, we're asked to determine the value
of F inverse of five without
finding F inverse of X.
And then, for part B, we're
asked to determine the value
of the derivative of F
inverse at X equals five.
So, looking at our notes below,
if F of A equals B and
F of X is differentiable
at the point A comma B, and F
perm of A doesn't equal zero,
then F inverse of B equals A
and F inverse is differentiable
at the point B comma A
and the derivative of
F inverse at X equals B
equals one over F prime of F inverse of B
which also equals one over F prime of A
because A is equal to F
inverse of B as stated here.
So, you can use this theorem
to answer part A and B.
To answer part A, we use the fact
that if F of A equals B,
then F inverse of B equals A.
So, if we can find an X value
where F of X would be equal to five,
we can then determine F inverse of five.
So, let's actually make a table of values
where we'll let the first column be X
and the second column be F of X.
Again, we're looking for an X value
where F of X would be equal to five.
Well, if we let zero equal zero,
we can easily see that F of
zero is not going to be five.
It would be zero plus
zero plus one, or one.
So, because F of zero equals one,
this does tell us that F
inverse of one equals zero.
But again, we're trying
to find F inverse of five.
So, if we try X equals one,
we can probably easily
see this is not going
to give us a function value of five.
F of one would be equal to,
this would be 1/4 times
one cubed which is 1/4
plus one plus one.
That would be 2 1/4 or 9/4.
So, because F of one equals 9/4,
this does tell us that F
inverse of 9/4 equals one.
But again, we're looking
for F inverse of five.
Let's try X equals two.
Let's show some work on this one.
F of two is equal to 1/4 times
two cubed plus two plus one.
Well, this would be 1/4
times A which is two.
Two plus two plus one is equal to five,
so we found the value we needed.
F of two is equal to five
which also tells us that F inverse of five
is equal to two.
So, for part A, we'll say
since F of two equals five,
we know F inverse of five must equal two.
And now, to find the
value of the derivative
of F inverse of F equals five,
we'll use our formula here
where the derivative of F
inverse at X equals five
is equal to one over F
prime of F inverse of five.
But, now we know that if
inverse of five is equal to two,
so this derivative function value
is equal to one over F prime of two.
So, before we find this though,
let's look at what this means graphically.
The given function F of
X is graphed here in red.
The line Y equals X is
graphed here in black.
So, to graph the inverse function,
we can reflect the red
function across Y equals X
which gives us this purple function.
So, this purple function
is F inverse of X.
And for part A, we found this point
on the original function, two comma five,
and from here, we found
the corresponding point
on the inverse function which
was the point five comma two.
So, this point represents
the function value
F inverse of five equals two.
And now, the formula for the
derivative of F inverse of X
is telling us that the
slope at the tangent line
at this point five comma two
is equal to the reciprocal
of the slope of the tangent
line at the point two comma five
to the original function.
And that's what this derivative
function value is here.
This is the reciprocal of
the slope of the tangent line
at the point two comma five
to the original function
which again, would give us
a slope at the tangent line
to the inverse function at X equals five.
For next step, we'll find F prime of X.
So, F prime of X is equal to,
well, here we'd multiply by three.
That'd be 3/4.
Subtract one from the exponent.
So, that's X squared.
The derivative of X is one.
The derivative of one is zero.
So, now we need to find F prime of two
which would be 3/4 times
two squared plus one.
So, here we're going
to have 3/4 times four.
That's just three.
Three plus one is equal to four.
So, for part B, we have
the derivative of F inverse
at X equals five equals 1/4.
So, going back to our graph one last time.
We just found the slope
of this green tangent line
to the inverse function at
the point five comma two
which was 1/4.
And we found this by using the fact
that this slope was a reciprocal
to the slope of the tangent
line divisional function
at the point two comma five.
The slope of this green
tangent line is equal to four.
I hope you found this helpful.
