What is the restriction for the sine function?

[teng125]

can anybody pls give me an example of this f(f^-1) = x??
thanx...

 

[abszero]

It's the definition of the inverse of a function f(x). For example, y = 2x and y^-1 = 1/2 x

 

[teng125]

how about if i want to write it using sin x??

 

[HallsofIvy]

Although I would say f(f^-1(x))= x. You're missing the "x" on the left side!

"arcsine" is defined as the inverse of sine (that's why your calculator has them paired). sin(arcsin(x))= x.

There are "technical" problems. Since sin(x) is not "one-to-one" (sin(\pi)= 0= sin(0)) there can't be a true inverse (a function can't return both 0 and \pi for x= 0). What is normally done is restrict the sine function to x=0 to \pi (which is really a different function than sine defined for all x) so that arcsin returns the "principal value"- the value between 0 and \pi.

 

[VietDao29]

What is normally done is restrict the sine function to x=0 to \pi (which is really a different function than sine defined for all x) so that arcsin returns the "principal value"- the value between 0 and \pi.

We don't really restrict sin function to x = 0 to \pi.
We, however, restrict sin function to x = -\frac{\pi}{2} to x = \frac{\pi}{2}. :)

 

[HallsofIvy]

We don't really restrict sin function to x = 0 to \pi.
We, however, restrict sin function to x = -\frac{\pi}{2} to x = \frac{\pi}{2}. :)

Oops! It's cosine that is restricted to "between 0 and \pi!