Question

How do you find the domain and range of `p(x)=-x^3 - x^2 + x -10`?

Answer 1

Domain: all real numbers
Range: all real numbers

Explanation:

...The domain of the function is the set of all numbers x such that f(x) is defined.

Typically, you look for function definitions involving a division, in which case you look for the devisor - any value(s) of x that would result in a division by zero is disallowed.

For real number analysis, (which you are definitely doing in algebra) you can look for function definitions that include a square root operation. A value of x that would result in the square root of a negative number would be excluded.

Neither of these applies to this function. x can take on any value and the function is still well defined. Therefore, the domain of the function is all real numbers.

The range of the function is the set of all values that f(x) can take.

For this function, your highers power term, `-x^3`, can take on negative values when x is positive, and it will take on positive values when x is negative. As the higher power goes, so goes the rest of the function, so, therefore, f(x) can take on any value, positive or negative. And so you can say that the range of the function is all real numbers.

...always helpful to have a graph of the function as a sanity check:
graph{-x^3-x^2+x-10 [-42.57, 49.86, -36.26, 10]}

GOOD LUCK