- #1
synkk
- 216
- 0
prove
whether or not the following functions are surjective or injective:
1) g: \mathbb{R} \rightarrow \mathbb{R} g(x) = 3x^3 - 2x
2)g: \mathbb{Z} \rightarrow \mathbb{Z} g(x) = 3x^3 - 2xmy working for 1):
injective: suppose g(x') = g(x) : 3x'^3 - 2x' = 3x^3 - 2x this does not imply x = x' hence not injective
surjective: need to show \forall y \in \mathbb{R} \exists x \in \mathbb{R} s.t. g(x) = y, y = 3x^3 - 2x as cubics always have one real root then as ## y \in \mathbb{R} ## ## \exists x \in g(x) \in \mathbb{R} ## s.t. ## g(x) = y ## therefore it's surjective
2):
injective: same as 1:
surjective: I'm not sure how to phrase this for the integers,
overall I'm not happy with my proofs, for the injectivity I haven't really shown that x is not equal to x', how would I do it? And for surjectivity I have mainly written it in words, how would I write it out formally for both questions?
whether or not the following functions are surjective or injective:
1) g: \mathbb{R} \rightarrow \mathbb{R} g(x) = 3x^3 - 2x
2)g: \mathbb{Z} \rightarrow \mathbb{Z} g(x) = 3x^3 - 2xmy working for 1):
injective: suppose g(x') = g(x) : 3x'^3 - 2x' = 3x^3 - 2x this does not imply x = x' hence not injective
surjective: need to show \forall y \in \mathbb{R} \exists x \in \mathbb{R} s.t. g(x) = y, y = 3x^3 - 2x as cubics always have one real root then as ## y \in \mathbb{R} ## ## \exists x \in g(x) \in \mathbb{R} ## s.t. ## g(x) = y ## therefore it's surjective
2):
injective: same as 1:
surjective: I'm not sure how to phrase this for the integers,
overall I'm not happy with my proofs, for the injectivity I haven't really shown that x is not equal to x', how would I do it? And for surjectivity I have mainly written it in words, how would I write it out formally for both questions?