Problem 3 — A Complete Introduction to Abstract Algebra

Let the functions

\displaystyle f:[0,\infty)\to\mathbb{R},\;x\mapsto x^{2}+1,

\displaystyle g:(-1,1]\to\mathbb{R},\;x\mapsto 1-x^{2},\text{ and}

\displaystyle h:(-\infty,-1]\to\mathbb{R},\;x\mapsto\ln(-x).

Which of the given function(s) are injective? If they are, prove it. If not, explain why.
Does the composite function $hf$ exist? If so, give a definition of $hf$ using arrow notation and state its image. Otherwise, explain why.
Does the composite function $fg$ exist? If so, give a definition of $fg$ using arrow notation and state its image. Otherwise, explain why.

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