# If you are given a set of points on a graph (0,0), (1,4), (2,1), (3,3), (4,5), how do you determine the domain of the function?

May 19, 2016

The domain contains the set $\left\{0 , 1 , 2 , 3 , 4\right\}$

#### Explanation:

The domain of a function is the set of values which the function can operate on. In this case, that means the set of possible $x$ values. Without knowing anything about the function beyond the given points, we cannot say with certainty which $x$ values are valid except for the ones already shown, that is, the values $0 , 1 , 2 , 3 ,$ and $4$.

For example, the function could be defined as a function $f$ such that
$f \left(0\right) = 0$
$f \left(1\right) = 4$
$f \left(2\right) = 1$
$f \left(3\right) = 3$
$f \left(4\right) = 5$

As $f \left(x\right)$ is only defined for $x \in \left\{0 , 1 , 2 , 3 , 4 , 5\right\}$, those would make up its entire domain.

Alternatively, we could have a function like

g(x) = −17/24x^4+25/4x^3−415/24x^2+63/4x

At the given points, $g \left(x\right)$ is identical to $f \left(x\right)$, however $g \left(x\right)$ is defined for all real numbers, and thus the domain for $g \left(x\right)$ is $\mathbb{R}$.

As we can see, there is no way to know the exact domain without knowing more about the function. We can only say that it contains the given values.