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

Sep 2, 2015

Domain and range are both $\setminus m a t h \boldsymbol{R}$.

#### 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 $f \left(x\right)$ with odd degree (in your case the degree is 3), has the following properties:

1. $\setminus {\lim}_{x \setminus \to - \setminus \infty} f \left(x\right) = - \setminus \infty$
2. $\setminus {\lim}_{x \setminus \to + \setminus \infty} f \left(x\right) = + \setminus \infty$

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