To find the image of the point \((3, 0)\) after a dilation centered at the origin with a scale factor of \(\frac{1}{3}\), you can use the following formula for dilation:
\[ (x', y') = (kx, ky) \]
where \((x, y)\) is the original point, \(k\) is the scale factor, and \((x', y')\) is the image after dilation.
In this case, the original point is \((3, 0)\) and the scale factor \(k\) is \(\frac{1}{3}\).
Now, apply the formula:
\[ x' = \frac{1}{3} \cdot 3 = 1 \] \[ y' = \frac{1}{3} \cdot 0 = 0 \]
Therefore, after the dilation, the image of the point \((3, 0)\) is
\[ (1, 0) \]