What is the domain and range (interval notation) for x^3-3x^2+4?

Question

I need help figuring this out. I'm working on a project for math and unfortunately I don't have the best teacher. I think its negative infinity to positive infinity, but I'm not sure. 

Answer 1

You are correct.  

 

x^3 - 3x^2 + 4 is a polynomial.  The domain of any polynomial is (-∞,∞), because you can put in any real number for x (meaning there are no restrictions on the domain).  To figure out the range you need to consider the end behavior of the function - this is what happens when x gets really big positive (towards +∞) or really big negative (towards -∞).  For a polynomial this behavior is determined by the leading term of highest degree - which in this case is x^3.  When x is really big positive then x^3 will be really, really big positive and it will be a lot bigger than any of the other terms in the polynomial so the y value will go towards +∞.  When x is really big negative then x^3 will be really, really big negative and it will be a lot bigger negative than any of the other terms in the polynomial so the y value will go towards -∞.  Since a polynomial is continuous it will take on every value between these 2 extremes so the range is (-∞,∞).