If f and g are one-to-one functions defined on all of R , then the function h(x)=f(x)+g(x) is also one-to-one. True or false and why?

Answer 1

FALSE

Example:

Let f(x) = x³ and g(x) = 1 - x³

f and g are both one-to-one by the Horizontal Line Test

But, h(x) = f(x) + g(x) = 1 for all x.

The graph of h is a horizontal line. So, by the Horizontal Line Test, h(x) is not one-to-one.

Answer 2

Suppose f(x)=x² and g(x)=-x² for x>0

Both f and g are 1-to-1 for x>0

h(x)=0 identically for x> and is NOT 1-to-1 for x>0