Inverse Functions

[jinx24]

Hi all...I'm back.

"Let f(x) = (2x - 3)/(x + 4). Find f^-1(x). Find the domain and range of f(x) and f^-1(x). What do you observe?"

I got all the way down to where it asks "what do you observe" I see what I think I am supposed to see, but I don't know how to put it into words.

f^-1(x) = (-4x - 3)/(x -2)

Domain f(x) = all real numbers except -4
Range f(x) = 3/2

Domain f^-1(x) = all real numbers except 2
Range f^-1(x) = -(3/4)
I see that the numbers in the domain and range in one function are in the equation of the opposite function. Is there a more mathematical way of saying this? Or even a better way?

One more question:
"assume that the domain of f and f^-1 is all real numbers. Solve the equation for x using the given information."

f^-1(2x + 3) = 5 ; f(5) = 13

I'm not sure what they are asking me to do. I know I can't plug the 5 into the inverse function because then I would not be solving for x. How do I get started with this?

Thank you so much.
Jenny

 

[Daniel_Feldman]

Hi there.

So we've got this function:

f(x) = (2x - 3)/(x + 4).

Its domain is of course (-inf,-4) U (-4,inf).

The grap has a vertical asymptote at x=-4. If you look at the end behavior of the function, you'll notice that as x approaches -4 from the left, f(x) approaches infinity. As x approaches -4 from the right, f(x) approaches -infinity. So at first glance it might appear that the range is all real numbers. However, this graph has a horizontal asymptote (y=2), so its range is (-inf,2) U (2,inf). I'm not quite sure how you got 3/2 there. The range is the set of all possible y-values.


Now, we get to the really important part. Notice your answer for the domain of the inverse function. How does it relate to my answer above for the range of f(x). Can you draw any conclusions from there about inverse functions (especially between their respective domains and ranges-that's what the "what do you observe" question is asking)?

If you manage to draw the conclusion, your second problem becomes MUCH easier.

 

[jinx24]

Thank you, Daniel. I finally at the point where I can determine the Domain of a function...finding the range is another story. I just glanced at an old book and saw that to find the range, the numerator cannot be 0. I probably should have read on!

I graphed the basic function and figured out where the horizontal asymptotes would be by (a/c). So, just like you said, the range for the original function is (-inf, 2) U (2, inf) and the range for the inverse is (-inf, -4) U (-4, inf). So the range for the inverse function is the same as the domain of the original function. Same with the range of the original function...its the domain of the inverse.

Just wanted to say thanks.
Jenny