# Domain/Range for Inv. Functions

## Homework Statement

Determine if each of the following is true or false. If false, give a counterexample.

Arccot (cot x) = x for all x

## Homework Equations

$$cot= 1/tan x$$
Principal values of tan are -90 degrees <x <90 degrees
Capitalization of trig function indicates a function with restricted domain

## The Attempt at a Solution

I figured since the cot = 1/tan, it would still have an infinite domain and range.
By that same thinking, the Cot would have the same domain and range as the Tan, -90 degrees < x <90 degrees.

But, we dont have Cot, we have the inverse of the Cot. So, do I just switch the x and y values?
I.e., the x values become infinity, and the y values become -90 degrees < x <90 degrees?

If so, does this make the equation true?

Mark44
Mentor

## Homework Statement

Determine if each of the following is true or false. If false, give a counterexample.

Arccot (cot x) = x for all x

## Homework Equations

$$cot= 1/tan x$$
Principal values of tan are -90 degrees <x <90 degrees
Capitalization of trig function indicates a function with restricted domain

## The Attempt at a Solution

I figured since the cot = 1/tan, it would still have an infinite domain and range.
But tan(x) is not defined for all values of x, and cot(x) is undefined at values of x for which tan(x) = 0.
By that same thinking, the Cot would have the same domain and range as the Tan, -90 degrees < x <90 degrees.
No, Cot and Tan don't have the same domains.
But, we dont have Cot, we have the inverse of the Cot. So, do I just switch the x and y values?
I.e., the x values become infinity, and the y values become -90 degrees < x <90 degrees?

If so, does this make the equation true?

Here's a similar example: Arcsin(sin(pi)) = 0. This shows a value x for which Arcsin(sin(x)) $\neq$ x.

Ok, I must really be confused.
Can you walk me through your thinking when determining whether the equation was true or false?

Here's a similar example: Arcsin(sin(pi)) = 0. This shows a value x for which Arcsin(sin(x)) $\neq$ x.

I dont understand what your saying here. Your example does not look like the equation of the problem. And, by my understanding, the arcsin and sin cancel, leaving you with pi, not zero.

Mark44
Mentor
Arcsin and sin "cancel" only under certain circumstances. You shouldn't even be thinking about cancelling, since that term refers to eliminating common factors in the numerator and denominator of a fraction or rational expression.

sin(pi) = 0, and arcsin(0) = 0, so for this value, pi, clearly arcsin(sin(pi)) $\neq$ pi.

The example I gave is similar to the one you're working. The similarity is that it is working with trig and inverse trig functions. My example shows that there are some subtleties at play here, involving domain and range.

Speaking of which, you have some misconceptions about the domain and range of the functions you're working with, as noted in my previous post.

sin(pi) = 0, and arcsin(0) = 0, so for this value, pi, clearly arcsin(sin(pi)) $\neq$ pi.

The example I gave is similar to the one you're working. The similarity is that it is working with trig and inverse trig functions. My example shows that there are some subtleties at play here, involving domain and range.

Ok, I understand your example now.

However, I still dont know how to determine whether the equation is true or false. The cot x is defined for 0< x <180, and y values being infinite.

Now, to find the inverse, you just exchange the x and y values with each other, right? So, x becomes infinite and y becomes 0<y<180.

But how does this equate to Arccot (cot x) = x for all values x?

vela
Staff Emeritus
Homework Helper
Ok, I understand your example now.

However, I still dont know how to determine whether the equation is true or false. The cot x is defined for 0< x <180, and y values being infinite.
This isn't true. The cotangent function is defined for all real numbers except multiples of pi.

This isn't true. The cotangent function is defined for all real numbers except multiples of pi.

Ok, I guess I can see that. But, I still dont see how to go from knowing that to knowing whether or not the equation is true or false.

eumyang
Homework Helper
Now, to find the inverse, you just exchange the x and y values with each other, right? So, x becomes infinite and y becomes 0<y<180.
No, not for trig functions. For instance:
f(x) = sin x
The domain is all reals, and the range is [-1, 1]

The inverse is the arcsin:
g(x) = arcsin x
BUT while the domain is [-1, 1], the range is NOT all reals. The reason is that you wouldn't have a function at all -- you would have multiple output values for a single input value. So we usually say that the range for g(x) = arcsin x is restricted to [-pi/2, pi/2].

Given vela's correction, you'll have to do something similar for f(x) = cot x and g(x) = arccot x.

69

vela
Staff Emeritus