Range of nonlinear function

$\begingroup$

I'd like to know the range of function $$f(x)=(x^2, x^3)$$ and how to proceed in these problems.

(The domain is $R$ while the codomain is $R^2$.)

What is the image of interval $[0,1]$?

$\endgroup$ 6

2 Answers

$\begingroup$

Given a function $f:A\to B$ (this notation means $A$ is the domain and $B$ is the codomain of the function), the range of $f$ is defined as:

$$Range(f)=f(A) = \{b\in B~:~\exists a\in A~\text{such that}~f(a)=b\}$$

In words, the range of $f$ is the set of all elements in the codomain which have a preimage, i.e. some element in the domain, which maps to it.

Some times we can simplify things further, but in this case, the way to write the range is going to look very similar to the basic definition.

For your specific example, we have

$$Range(f)=\{(x^2,x^3)\in\Bbb{R}~:~x\in\Bbb R\}$$

For the range of the specific interval, you have:

$$f([0,1])=\{(x^2,x^3)\in\Bbb{R}~:~x\in[0,1]\}$$

$\endgroup$ 1 $\begingroup$

This is a parametrisation of the cubic curve with equation $y^2=x^3$, i.e. $y=\pm\sqrt{x^3}$. Here is how it looks like:

$\endgroup$ 2