R and g: RR
by f(x) = | x | + 1 and g(x) = 2x3-1, for all x in R.
2. Define the functions f: RR and g: RR
by f(x) = | x | + 1 and g(x) = 2x3-1, for all x in R.
The function f is not one-to-one since f(-2) = |-2| + 1 = 3 and f(2) = |2| + 1 = 3, but -2 is not equal to 2.
The function g is one-to-one. Suppose there exist real numbers x1 and x2 such that g(x1) = g(x2). Then,
| 2(x1)3 - 1 | = | 2(x2)3 - 1 |
| 2(x1)3 | = | 2(x2)3 |
| (x1)3 | = | (x2)3 |
| x1 | = | x2. |