Proving a cubic is surjective.

[pondzo]

Homework Statement

$$f:\mathbb{R}\rightarrow\mathbb{R}~~\text{where}~~f(x)=x^3+2x^2-x+1$$
Show if f is injective, surjective or bijective.

Homework Equations

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 I am 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?

Thank you for your time.

 

[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.

 

[Dick]

Homework Statement

$$f:\mathbb{R}\rightarrow\mathbb{R}~~\text{where}~~f(x)=x^3+2x^2-x+1$$
Show if f is injective, surjective or bijective.

Homework Equations

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 I am 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?

Thank you for your time.

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