Determine which of the following functions are onto.

Question

A) f: R → R defined by f(x) = x³ + x.

B) f: R³→R³ defined by f(x, y, z) = (x − y, yz, xz).

C) f: R → R defined by f(x) = x².

D) f: R → R defined by f(x) = x³.

E) f: R³→ R³ defined by f(x, y, z) = (x + y, y+z, x+2).

Answer 1

The term "onto", or surjectivity, means every possible output can be achieved (basically a range from -inf to inf)

A) f(x) has a range of -inf to inf, so it is onto

C) f(x) can never be negative, so it is not onto

D) same as A

B) By counterexample: (1,1,0) is not a valid result vector, so its not onto

E) Since the third term is "x-2", it is not a linear transformation. So another way to prove "onto" is if result vector (a,b,c) can be expressed in terms of x,y,z:

x = c-2

y = a-c+2

z = b-a+c-2

So it is onto