calculus ?

Question

If f(x) is one-to-one, can anything be said about g(x)= -f(x)? Is it also one-to-one? Give reasons.

Answer 1

Assume f(x) is one-to-one. Then it follows that, by definition,

if f(a) = f(b), then a = b.

Let g(x) = -f(x). Then

g(a) = -f(a), and
g(b) = -f(b).

Assume g(a) = g(b). Then
-f(a) = -f(b). Multiply both sides by (-1), we get
f(a) = f(b). Since f(x) is one-to-one, then
a = b.

Therefore, we've just shown that if g(a) = g(b), then a = b.
That makes g(x) one-to-one.

Answer 2

If they are both continuous functions and share the same domain that is the case.

Answer 3

If f(x) is one-to-one, then multiplying it by -1 should not change that characteristic. (Now, if you multiplied it by a variable, that could change whether it is one-to-one.)