.determine whether the function f(x)^2-3(x)+1 is one-to-one, onto, or both. Prove your answer. The domain and codomain if f(x) is the set of all real numbers.
f(x) = 3x^2 -3x + 1
the function is onto Y, but not one-to-one:
because 1 is a real number and
f(1) = 3(1)^2-3(1)+1 = 1
and 0 is a real number
f(0) = 3(0)^2-3(0)+1 = 1
there exists two different elements in x that map to a single element in y; therefore, f is not one-to-one.
the function is onto Y, since all real numbers of Y coexist with all real numbers of X.
"Coexist"? That is not a mahtematical term. The function is not onto the real numbers, not even close: that quadratic's
discriminant is , and this means the function is always positive or always negative...
OK, my question is did I correctly prove that
1. the function is one-to-one?
2. how do I prove the last one is true algebraically?
I assume its true since the domain is (-00, 00) and range is(-00, 00)?