Give an example to show that the product of two one-to-one function is not necessarily a one-to-one function.
y=x
y=-x
y^2 = -x^2
Sorry I don't understand y^2=-x^2 ??
Shouldn't it be:(y)(y)=(x)(-x)
y^2 = -x ??
(x)(-x) = -x^2
To show that the product of two one-to-one functions is not necessarily a one-to-one function, let's consider the following example:
Function f: {1, 2, 3} → {2, 4, 6}
Definition: f(x) = 2x
Function g: {2, 4, 6} → {1, 2, 3}
Definition: g(y) = y/2
Both functions f and g are one-to-one functions as each input has a unique output. However, their product function is not a one-to-one function.
Now, let's find the product of these two functions:
h(x) = g(f(x))
= g(2x)
= (2x)/2
= x
As you can see, the product function h(x) is simply the identity function, where every input x maps to itself. While f and g are both one-to-one functions, their product h is not. This is because multiple inputs from the domain of f can map to the same output in g, resulting in a loss of distinctiveness in the product function.
Therefore, this example demonstrates that the product of two one-to-one functions is not necessarily a one-to-one function.