If it is not an onto function, provide the range.(adsbygoogle = window.adsbygoogle || []).push({});

I have problems with these two

g: R -> R

g(x) = x^2 + x

g(x) = x^3

The book said g1 is neither onto nor 1-1, while g2 is both

I see how x^2 + x is not `1-1 function. But how come it is not onto? I mean R is real number, can be 343.3434534564656757 or 67.23123132 or anything that is real number. How can we quickly show that x^2 + x is not onto? There are -inf, inf real numbers, so how can we say there is a particular g(x) left out?

One way I see is if x^2 + x = 0 and we get x = sqrt(-x) which is an i num

I see x^3 is a 1-1 because x1 = x2. in a similar way as in the case of g1, how can we be so sure that all real numbers can be produce?

In the book, things like x^3 its surjectiveness can be proven as following

x^3 and if r is any real number in codomain of f, then real number cubic root of r is in the domain of f and since f(cubic root of r) = (cubic root of r)^3 = r, so f = R = range of f, so it is onto (copied from book)

Thanks

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# Determine whether it is an onto function

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