1) Prove that there exists a set A such that f: R-->A, defined by f(x) = x^2 is NOT onto.
2) Let f:[-2,2] --->[0,4] be defined by f(x) = x^2.
Prove that f is onto.
Any help would be appreciated Thanks!
2) Let f:[-2,2] --->[0,4] be defined by f(x) = x^2.
Prove that f is onto.
Any help would be appreciated Thanks!
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1) Take A to be any interval in IR whose left end is to the left of 0. For example, A = (-1, ∞). Note that there are no real x such that f(x) = -1/2. The mapping is not onto A.
2) Let x in [0, 4] be arbitrary. Note that √(x) is well defined as a real number, 0 ≤ √(x) ≤ 2, and
f(√(x)) = (√(x))² = x.
As x was arbitrary, f is onto.
2) Let x in [0, 4] be arbitrary. Note that √(x) is well defined as a real number, 0 ≤ √(x) ≤ 2, and
f(√(x)) = (√(x))² = x.
As x was arbitrary, f is onto.