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Keil MDK is the most complete software development system for Arm-based microcontrollers and includes all components that you need to create, build,Q:
Prove that $f(x) = x^3 + x^2 + 1$ is a surjective function
How do I prove that $f(x) = x^3 + x^2 + 1$ is a surjective function?
I know that if $f: \mathbb{R} \rightarrow \mathbb{R}$, then I need to find $g: \mathbb{R} \rightarrow \mathbb{R}$ such that $g \circ f(x) = x \ \ \forall x \in \mathbb{R}$, which is $g(x) = x^3 + x^2 + 1$.
How do I prove $g$ is surjective? I mean that is it true that $f(\mathbb{R}) = \mathbb{R}$?
A:
Hint: Note that $f(\pm1)=f(-1)+f(1)=1+1+1=3=1$ and $f(0)=1+1+1=2$. Since $f(x)>0$ and $f(-x)=-f(x)$, you can show that $f(x)$ is surjective.
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The present invention generally relates to a hand-operated adhesive dispenser and, more particularly, to an integrated hand-operated dispenser for dispensing tissue and/or other adhesive material.
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