You might want to get away from trig functions exclusively and
talk about notation more generally, then return to trig.
I.E. you could say something like:
f(x) = y means I'm "doing something" to x and getting y. Now
if I want to get back to x, I might have an "undo function"
such that g(y) = x. If I have such a do/undo relationship
between two functions, I typically call one the "inverse" of
the other and signify this as follows:
If g(y)=x for any f(x)=y, then g = f^-1 and f = g^-1
Another way for writing it:
If f and g are inverse functions, then g(f(x))=x
Then comes some fine tuning about domain and range, how the
two functions might not be precisely inverses. For example
if f(x) = x^2 then g(y) = y^0.5 -- except 4^0.5 could be
-2 or 2 i.e. there's a "fork in the road" on the return
journey which makes g(y) a relation, not a function.
Many-to-one in one direction means one-to-many in the other.
I think what confuses students is that the notation is in
fact somewhat confusing. If f(x) = 1/x, then g(x) = 1/x is
an inverse (reciprocates reciprocation to return the original).
f(3)= 1/3 and g(1/3)= 3. f^-1 = g, but also g(x) = f(x) = x^-1.
Returning to trig, you could inform your students that many
computer languages use asin and acos for sin^-1 and cos^-1.
There's less temptation to misread asin as 1/sin.
asin(sin(x)) = x (stipulations about domain and range)
Your students' other confusion might be similarly addressed,
by getting away from trig notation to examine the more general
case. E.G.
If f(x) = x^2 then f(3) = 9,
but f(2*3) = 36 is not equal to
2*f(3) = 18
i.e. 2f(x) <> f(2x), just as 2sin(x) <> sin(2x)
Kirby
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