A function g:R-->R that is not onto and not 1-1
cos(x) is a function g:R->R so that f is not onto and f is not 1-1.
We know from calculus that cos x is a function. But, it is not so easy to rigorously prove - t ry it without assuming what you are trying to prove ! Let's assume that we have proved that cos x is a function.
To prove that cos x is not 1-1 we must produce x1 not equal to x2 so that cos(x1)=cos(x2).
Take x1=Pi and x2=3Pi. Notice that x1 and x2 are real and unequal. Also,
and so cos(x1)=cos(x2). Hence cosx is not 1-1.
To prove that cos x is not onto we must produce y so that cos(x) is not equal to y for all x in R.
Take y=2. Assume for contradiction that cos(x)=2 for some x in R.
Then x=1.316957897 I. We have arrived at a contradiction to the fact that x is in R, since x is complex.
Hence, cos(x) is not equal to 2 for all x in R, and so cos x is not onto.