Question

What is the domain and range of `f(x) = x^3 - 3x + 2`?

Answer 1

Domain and range are both `\mathbb{R}`.

Explanation:

The domain is defined as the set of the points which you can give as input to the function. Now, "illegal" operations are:

1. Dividing by zero
2. Giving negative numbers to an even root
3. Giving negative numbers, or zero, to a logarithm.

In your function, there are no denominators, roots or logarithms, so all values can be computed.

As for the range, you can observe that every polynomial `f(x)` with odd degree (in your case the degree is 3), has the following properties:

1. `\lim_{x \to -\infty} f(x)=-\infty`
2. `\lim_{x \to +\infty} f(x)= +\infty`

And since polynomials are continuous functions, the range consists in all numbers from `-\infty` to `\infty`, which is to say all the real set.