How Do You Find the Image of a Function Analytically?

[Iclaudius]

Hi all,

is there a way that one goes about solving for the image of a function in a purely analytical sense without having to use any sort of graphing calculator? btw I'm abit confused about the concept of image - it seems it is not the range which is what i thought it was :S? can someone please explain this to me or give me a link to a website resource perhaps?

I have been looking online and have not been able to find any good resources on image of a function - just a lot of things on range/domain.

Thanks in advance,
Claudius

 

[micromass]

Range and image are (often) used interchangeably. So any resources that you find on range, will probably be suitable for finding the image too.

The quick way to find the image of a function is by solving equations: Say you have a function f, then y is in the image of f if there is an x such that y=f(x). So all you need to do is decide whether there exists such an x or not.

Let's give two simple examples:
Let f(x)=1/x. Take an arbitrary y. If there exists an x such that y=f(x), then y=1/x. This must necessarily mean that x=1/y. But the last expression only makes sense if y is not zero. So, if y is not zero, then f(1/y)=y. Thus the image of f is everything but 0. A quick look at the graph will tell you thesame thing.

Another example. Let f(x)=x². Take an arbitrary y. If there exists an x such that y=f(x), then y=x². This means that $$x=\sqrt{x}$$. But this last expression only makes sense if x is nonnegative. Thus the image of f are all the nonnegative real numbers. Again, a look at the graph will confirm this.

 

[Iclaudius]

Thanks for the reply that cleared things up

 

[micromass]

Hey micro thanks again for the reply,

another quick question to check my comprehension - say we have a function,

F(x)= 2(x-2)^(2) / x^(2)+x-6

to find the image - all we need to do is find the inverse of the function.

then we plug in y for x, f(y)=y and see where the equation falls apart? so in this case i got y cannot equal 2, so was just wandering what notation do we use to express this?

thanks in advance,
Claudius

Yes, that is basically correct. But I have to be pedantic here: you cannot say that you calculate the inverse of the function, because the function has no inverse (indeed the inverse is not defined at 2). I know this is silly, and what you mean is correct, but I just wanted to make things clear!

And about the notation, there are some notations that you can use to represent your result. I, personally, would use $$Im(F)=\mathbb{R}\setminus\{2\}$$. But you will often see the notation $$]-\infty,2[\cup ]2,+\infty[$$, or even $$(-\infty,2)\cup (2,+\infty)$$. All three notations are correct, but maybe your teacher will want you to use a specific one, I don't know...