Question

d

Answer 1

You donʻt have any ideas yourself?

Answer 2

Even just a guess :)

Answer 3

so what is sin(3pi)=?

Answer 4

or sin(540)=? if you prefer degrees

Answer 5

Suppose, you have any function \(\color{#000000}{ \displaystyle y=f(x) }\)
with some domain \(\color{#000000}{ \displaystyle x\in[a,b] }\), and some
range \(\color{#000000}{ \displaystyle y\in[c,d] }\), and you are to take the
inverse of this function.

To find the inverse-function, you are swiping
the \(\color{#000000}{ y }\) and \(\color{#000000}{ x }\), and solving for \(\color{#000000}{ y }\) (in terms of \(\color{#000000}{ x }\)).

After swiping \(\color{#000000}{ x }\) and \(\color{#000000}{ y }\), wouldn't this mean that,
(1) \(\color{#000000}{ y }\) - is now the \(\color{#000000}{ x }\), \(\bf and,\)
(2) \(\color{#000000}{ x }\) - is now the \(\color{#000000}{ y }\) ?

That is why, the range and domain of the inverse
function are now going to be (changed):
\(\color{#000000}{ \displaystyle x\in[c,d] }\)
\(\color{#000000}{ \displaystyle y\in[a,b] }\)

Answer 6

we want the inverse to be a function

Answer 7

x^2 on R is not injective and thus has no inverse but x^2 on [0,infinity] does.

Answer 8

Ok, so you just said that both x=3pi and x=pi would have the same element of the range assigned to them. That means that the inverse will have at least two elements of its domain(the range of sine) which will assign to different values. In order to be a function, can we have that? No. So we restrict the values in order to make the inverse a function.