Verifying one-to-one Algebraically
There are two ways of verifying that a function f is one-to-one algebraically:
 Method 1: Pick two arbitrary non-equal elements in the domain and show that
they do not go to the same value in the range.
 Method 2: Pick one element in the range and show that it cannot go to two
different elements in the domain.
The way to disprove a function is one-to-one is to give a counter-example.
Example: Use Method 1 to show that f(x) = x3 is one-to-one.
Proof: Let a and b be two real numbers and suppose that a ≠ b. Then f(a) = a3
and f(b) = b3. Now suppose that f(a) = f(b). We will show that this gives rise to a
contradiction. Since f(a) = f(b) then
a3  b3
a3  b3  0
(a  b)( a 2  ab  b 2 )  0
Since (a 2  ab  b 2 ) is never zero then a – b = 0. Therefore a = b which is a
contradiction. Thus f(a) ≠ f(b), and therefore, f is one-to-one.
Example: Use Method 2 to show that f(x) = x3 is one-to-one.
Proof: Let y1 be a value in the range and suppose that a and b both satisfy f(a) = f(b) =
y1. Then a3 = b3 , and
a3  b3
a3  b3  0
(a  b)( a 2  ab  b 2 )  0
Thus a = b.
Notice that the two methods are similar, except that in the first method we derived a
contradiction, and in the second we verified that they are the same.
Example: Show that f(x) = x2 is not one-to-one.
Proof: f(-2) = f(2) = 4.