# Solutions to Problem Set 25

Math 310-01/02

11-17-2017

1. is defined by . Prove that f is injective directly, using the definition of "injective".

Suppose . Then

Therefore, f is injective.

2. is defined by . Prove that f is injective, using the derivative.

for all x, so f is increasing for all x. Therefore, f is injective.

3. is defined by . Prove that f is surjective directly, using the definition of "surjective".

Let . I need to find such that .

First, I'll work backwards to "guess" the answer:

Next, I'll check that my guess works:

Therefore, f is surjective.

4. is defined by . Prove that f is surjective using the Intermediate Value Theorem.

Let . Note that

Hence, there are numbers a and b such that

f is continuous, so by the Intermediate Value Theorem, there is an such that .

Therefore, f is surjective.

Adversity makes no impression upon a brave soul. - Ali Ibn Abi Talib

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