Determine Direct and Inverse Image f(E) and f^-1 (G)...

phillyolly

1. The problem statement, all variables and given/known data

Let f(x):=1/x^2, x not equal 0, x belongs R
a) Determine the direct image f(E) where E:= (x belongs R : 1<=x<=2)
b) Determine the inverse image f^(-1)(G) where G:= (x belongs R : 1<=x<=4)

2. Relevant equations



3. The attempt at a solution

A) Let f: R -> R be defined by f(x):=1/x^2. Then, the direct image of the set E:=(x:1<=x<=2) is the set f(E)=(y:1<=x<=1/4).
If G:= (y : 1<=x<=4), then the inverse image of G is the set f^-1 (G)=(x:

And here I don't quite understand how to find an inverse image. Please help.

lanedance

how about starting by finding the inverse map...

if you choose a reasonable domain f will be 1:1, so its inverse will exist such that f^(-1)(f(x)) = x

phillyolly

I have no idea what you are talking about, I am sorry....

phillyolly

What I have done:

f^(-1)(x)=x^2

This is the inverse formula. What should I do now?

lanedance

I prefer to keep the variables differnt for the function & its inverse as below, i find it makes thing easier, so let:

[itex] y = f(x) [/itex] is a function that maps from x to y

for question a), if the domain of x is [1,2], then the y is in the range [1/4,1] as you've found

now for question b) assuming f is 1:1 we can find its inverse function, lets call it g
[itex] x = f^{-1}(y) = g(y) [/itex] which maps y back to x

lanedance

now with all that in mind, starting with the function
[tex] y = f(x) = \frac{1}{x^2}[/tex]

you've said the inverse funtion is
[tex] x = g(y) = f^{-1}(y) = y^2[/tex]

lets check if it actually satisfies the inverse property
[tex] f^{-1}(f(x)) = g(f(x)) = (f(x))^2 = (\frac{1}{x^2})^2 = \frac{1}{x^4} \neq x [/tex]

so this is not infact the correct inverse function

lanedance

to find the correct inverse, start with
[tex] y = \frac{1}{x^2}[/tex]

now solve for x in terms of y, that gives you g

phillyolly

I am so lost with your last comment. I can only see that this formula is the same as in your previous post.
Please any clues?

phillyolly

OK, I think I got it. Please check if I am right,

Attached Thumbnails

 

lanedance

yep that looks good
[tex] y(x) = \frac{1}{x^2}[/tex]

so rearranging gives
[tex] x(y) = \sqrt{\frac{1}{y}}[/tex]

HallsofIvy

Everyone has been focusing on (B) but let me make an obvious point 1 is not "less than or equal to" 1/4! Also, it make no sense to say "y such that x has some property". What you should have is f(E)= {y: 1/4<= y<= 1}.

phillyolly

Hi! This is an old posting of mine. I am going back to this discussion. Is it right that for question b the answer is:

f^(-1)(G)={1/2<=x<=1}

? Thanks!

HallsofIvy

No, it's not. Notice that if x= -1/2, y= 1/(-1/2)^= 4. f is an even function so its graph is symmetric about the y-axis. lanedance made an error when he said the inverse function is given by [itex]x= 1/\sqrt{y}[/itex]. In fact, this function does not have an inverse!

In my first year in grad school, I had to present, before the class, a proof involving [itex]f^{-1}(A)[/itex] where A is a set. I did the whole problem assuming that f had an inverse- very embarassing!

The way you should have approached the problem is to look for x such that f(x)= 1/x^2= 1 and f(x)= 1/x^2= 4. The first equation has solutions 1 and -1, the second has solutions -1/2 and 1/2. Further, the derivative of f is -1/x^3 which is always negative between 1 and 2 and positive between -2 and -1. The fact that the function is decreasing on one interval and negative on the other tells you the domain is just the two intervals, [itex]1< x\le 2[/itex] and [itex]-2\le x< -1[/itex].

If, for some x in either of those intervals, the derivative were 0, you would have to consider the possibility that there is a turning point in the interval.

lanedance

this isn't very fresh in my head, but I'm not sure I understand

Unless I'm missing something f(x):=1/x^2 on 1<=x<=4 is monotone decreasing with f'(x) not zero anywhere so the inverse exist

HallsofIvy

But it is not x, in the original function, that is between 1 and 4. The problem was to find [itex]f^{-1}(G)[/itex] where G is [1, 4]. If we write the original function [itex]y= 1/x^2[/itex]. Then we are looking for x such that [itex]1\le y\le x[/itex].

Yes, the problem was written in terms of [itex]f^{-1}(x)[/itex] with [itex]1\le x\le 4[/itex] but in terms of the original function, y= f(x), it is y that is between 1 and 4.

Is there a reason why LaTeX is not working?