# Proving a cubic is surjective.

1. Mar 15, 2015

### pondzo

1. The problem statement, all variables and given/known data
$$f:\mathbb{R}\rightarrow\mathbb{R}~~\text{where}~~f(x)=x^3+2x^2-x+1$$
Show if f is injective, surjective or bijective.
2. Relevant equations

3. The attempt at a solution

f is obviously not injective (and thus not bijective), one counter example is x=-1 and x=1.

I can see from the graph of the function that f is surjective since each element of its range is covered.
But im not sure how i can formally write it down. If i try this;

$\text{let}~y\in\mathbb{R}$ and then try to find which $x\in\mathbb{R}$ maps to y then the algebra is going to get really messy and it would take a long while. Is there a faster and easier way of showing f is injective? Can i draw the function and show that all of the co-domain is covered, or is this not formal enough?

2. Mar 15, 2015

### RUber

I think that stating that the function is continuous and tends toward plus or minus infinity for large arguments should be sufficient. You might need to put a little more math and logic into it, but that is the simple argument.

3. Mar 15, 2015

### Dick

Use the Intermediate Value Theorem. http://en.wikipedia.org/wiki/Intermediate_value_theorem