Help with inverse function.

[mhidk]

Hello to all
First time poster, I guess I'll be here a lot from now on.

I have a problem with finding an inverse function, it is technically a pre-cal question since it is in the first chap of calculus.

Anyway, I need to find the inverse of the function: f(x)=sqrt(x² -4x)

The back of the book says the answer is: f(x)^-1= 2 + sqrt(x² + 4x)

I am at a loss on how to come to this result. I appreciate any help. Thanks!

 

[Ishuda]

Hello to all
First time poster, I guess I'll be here a lot from now on.

I have a problem with finding an inverse function, it is technically a pre-cal question since it is in the first chap of calculus.

Anyway, I need to find the inverse of the function: f(x)=sqrt(x² -4x)

The back of the book says the answer is: f(x)^-1= 2 + sqrt(x² + 4x)

I am at a loss on how to come to this result. I appreciate any help. Thanks!

An inverse function is one which gives the starting input for the composite function, that is if g is the inverse of f then
g(f(x)) = x
and
f(g(x))=x

So let g(x) be the inverse function of
f(x) = (x²-4x)^1/2
That is
g((x²-4x)^1/2) = x
So what do you have to do to (x²-4x)^1/2 to get x?

BTW: If you have copied everything correctly, I think the book is wrong.
EDIT: Make that wrong (slightly) instead of just wrong. Also the page
http://coolmath.com/algebra/16-inverse-functions/05-how-to-find-the-inverse-of-a-function-01.htm
might be interesting to you.

 

[Steven G]

Hello to all
First time poster, I guess I'll be here a lot from now on.

I have a problem with finding an inverse function, it is technically a pre-cal question since it is in the first chap of calculus.

Anyway, I need to find the inverse of the function: f(x)=sqrt(x² -4x)

The back of the book says the answer is: f(x)^-1= 2 + sqrt(x² + 4x)

I am at a loss on how to come to this result. I appreciate any help. Thanks!

y =sqrt(x² -4x)
So x = sqrt(y² -4y). What constraints are there on y???
x^{2 =}y² -4y. This is a quadratic equation in y. So what do you do next?

I agree with Ishuda that the answer in the book is not complete.

 

[mhidk]

y =sqrt(x² -4x)
So x = sqrt(y² -4y). What constraints are there on y???
x^{2 =}y² -4y. This is a quadratic equation in y. So what do you do next?

I agree with Ishuda that the answer in the book is not complete.

Sorry I forgot to mention the constraint of [4, infin]

 

[mhidk]

Let me right the whole question since what I originally asked was just a part of a question.

a) sketch a graph of the function f
b) determine the interval on which f is one-to-one
c)find the inverse function f on the interval found in part (b)

function: f(x) = sqrt(x² -4x)

a)already done
b) [4, infint]
C)

I need c

 

[Steven G]

Let me right the whole question since what I originally asked was just a part of a question.

a) sketch a graph of the function f
b) determine the interval on which f is one-to-one
c)find the inverse function f on the interval found in part (b)

function: f(x) = sqrt(x² -4x)

a)already done
b) [4, infint]
C)

I need c

Not sure why x<0 but maybe it is not 1-1 there. In any case I gave you the next step to do c. Just use the quadratic formula to solve the quadratic equation x^{2 =}y² -4y for y. As a calculus student I can only assume that you know how to find the solutions to a quadratic formula. Just remember that you have a constraint on y!
Let us know how you make out.

 

[Ishuda]

Let me right the whole question since what I originally asked was just a part of a question.

a) sketch a graph of the function f
b) determine the interval on which f is one-to-one
c)find the inverse function f on the interval found in part (b)

function: f(x) = sqrt(x² -4x)

a)already done
b) [4, infint]
C)

I need c

Just as a matter of notation, your answer to b) indicates that the point 4 and \(\displaystyle \infty\) [infint] are included in the interval. That is the square bracket, [ or ], says include the point and the open bracket, ( or ), says don't include the point as a matter of convention. Now, according to some, I am the last one who should be talking about following convention, but, never the less, since \(\displaystyle \infty\) is not a real number, it should never (when talking about the Reals) be included in the set. As far as the 4 goes, what is f(4) and f^-1(f(4)) where f^-1 is given above in your initial post. The latter should be 4 if f^-1 is the inverse of f. After answering those questions, you might agree that we should have
b) (4, \(\displaystyle \infty\))


EDIT: Just noticed that you are supposed to define the interval, so, not only should the one interval be (4, \(\displaystyle \infty\)) but there is another set missing from your list. Have you considered (-\(\displaystyle \infty\),0] U {4} which gives rise to previous comments about solution not being complete.