Graphing Inverse Trigonometric Functions
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Key Questions

Since the graphs of
#f(x)# and#f'(x)# are symmetric about the line#y=x# , start with the graph of a trigonometric function with an appropriate restricted domain, then reflect it about the line#y=x# .(Caution: Their domains must be restricted to an appropriate interval so that their inverses exist.)
Let us sketch the graph of
#y=sin^{1}x# .The graph of
#y=sinx# on#[pi/2, pi/2]# looks like:By reflecting the graph above about the line
#y=x# ,The curve in purple is the graph of
#y=sin^{1}x# .The graphs of other inverse trigonometric functions can be obtained similarly.
I hope that this was helpful.

Throughout the following answer, I will assume that you are asking about trigonometry restricted to real numbers.
Using Domain of
#arc sin x# Find
#arc sin (3)# .#3# is not in the domain of#arc sin# , (#3# is not in the range of#sin# ) so#arc sin (3)# does not exist.Using Range of
#arc sin x# Find
#arc sin (1/2)# .Although there are infinitely many
#t# with#sin t = 1/2# , the range of#arc sin# restricts the value to those#t# with#(pi)/2 <= t <= pi/2# , So the value we want is#pi/6# .Using the quadrant
This is the same as using the range, but it involves thinking about the problem more geometrically.Find
#arc sin (1/2)# .#arc sin (1/2)# is a number (an angle) in quadrant I or IV. It is the#t# with smallest absolute value (the shortest path from the initial side).Again,
#arc sin (1/2) = pi/6# . 
For a trig function, the range is called "Period"
For example, the function
#f(x) = cos x# has a period of#2pi# ; the function#f(x) = tan x# has a period of#pi# . Solving or graphing a trig function must cover a whole period.The range depends on each specific trig function.
For example, the inverse function#f(x) = 1/(cos x) = sec x# has as period#2pi# .Its range varies from (+infinity) to Minimum
#1# then back to (+infinity), between (#pi/2# and#pi/2# ).Its range also varies from (infinity) to Max 1 then back to to (infinity), between (
#pi/2# and#3pi/2# ). 
It is best to start by graphing the parent inverse trig function. For example, if you graph the inverse tangent graph, you would have horizontal asymptotes where tangent has vertical asymptotes  which is where cosine is zero the first time in both directions  which is at  pi/2 and pi/2. You graph your horizontal asymptotes, plot the origin (because the tangent of zero is zero), then draw what looks pretty much like an
#x^3# graphing laying on its side.Apply the transformations to the horizontal asymptotes and to the point at the origin, and draw the transformed graph the same way.