can someone help me find inverse of the function?

[MATH]f^{-1}\left(f(x)\right) = x[/MATH]
What simple function of [MATH]5x[/MATH] would give you [MATH]x[/MATH] ?
 
Inverses are things you do with a calculator. Imagine after a long time of making calculations you have a number in your calculator and by mistake you multiply this number by 5. Now you don't remember the number which you multiplied by 5. You have two ways of obtaining that number again. You can start from the beginning of the calculations you need to make OR you can think of a better way to obtain that number which you multiplied by 5. How do you undo multiplying by 5? You divide by 5. You take this unknown number, which we will call x, and divide it by 5. That is the inverse function is x/5.

Sure you can do this in a mechanical way like Subhotosh Kahn suggested or you can think about it which is easier and fun.

Here are some examples:
The inverse of x+2 is x-2, the inverse of x-7 is x+7, the inverse of x/3 is 3x. Do you see the pattern?
 
Jomo said:
Inverses are things you do with a calculator. Imagine after a long time of making calculations you have a number in your calculator and by mistake you multiply this number by 5. Now you don't remember the number which you multiplied by 5. You have two ways of obtaining that number again. You can start from the beginning of the calculations you need to make OR you can think of a better way to obtain that number which you multiplied by 5. How do you undo multiplying by 5? You divide by 5. You take this unknown number, which we will call x, and divide it by 5. That is the inverse function is x/5.

Sure you can do this in a mechanical way like Subhotosh Kahn suggested or you can think about it which is easier and fun.

Here are some examples:
The inverse of x+2 is x-2, the inverse of x-7 is x+7, the inverse of x/3 is 3x. Do you see the pattern?
Click to expand...
"a mechanical way like Subhotosh Kahn suggested...."

But "mechanical way" will work every time (if inverse function can be expressed explicitly), e.g.

y = ln(a*x^2) → x = ln(a*y^2) → y = ?

All "jabbing" aside - I agree with Jomo (!!!) - there is no substitute for "thinking". (..."you can think about it"...)
 
Yes, the mechanical way always works but it is quite possible that a student can be quite good at finding inverses but does not understand what an inverse is at any level at all.
This is dangerous and is one of the reasons why students do not like math.
 
What is I think getting lost here is whether there is any real utility in teaching students how to find the inverse of a function.

Option A: There is broad class of functions for which an algorithm for finding there inverses is known, has been safely recorded for posterity, and has been rendered into freely available software. Assuming this class of functions meets the practical needs of 99% of the educated public, we should not torture kids with learning how to find inverses by hand (just as we should not torture them with learning how to add, subtract, multiply, and divide when it would be much easier to teach them how to use a calculator). The true purpose of modern education in math should be to teach how to determine which computer tools apply to problems. That is, the only thing to teach are word problems.

Option B: We should teach mechanics because, done properly, it helps the student understand when a technique is appropriate. But that implies that we should teach the mechanics in a way that meets that goal.

Option C: We teach the mechanics of math to teach disciplined, careful, logical thought.

I admit the plausibility of Option A, but feel uncomfortable with it for reasons I can’t quite articulate. I‘d reject Option C if there are other, more productive, ways to teach mental discipline in fields that computers cannot yet do. With respect to Option B, I’d adopt it if I thought we taught mechanics in a way that led to understanding. I doubt that saying “swap x and y in a formula and solve for y“ teaches what an inverse does and when it is relevant.
 
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