What is the functiong of a plant hormone?

It depends on how you look at it. However maybe we can say it is a messenger or "control packet" where a cell or a tissue tells another cell or tissue, for instance on the opposite side of the plant, to do something to fulfill a need or begin another part of the plant life cycle.

For instance, it has been my speculation that gibberellin is made by cells when they are experiencing deficiencies in sugar. So roots that are experiencing sugar shortages, might make gibberellin.

The various things that gibberellin does, are attempts to fix this problem. So gibberellin stops root growth because roots don't make sugar they just use it; gibberellin causes the release and dissolving of stored starch in cells; it causes the shoot to lengthen, presumably because the plant thinks it's being shaded; and it stops the senescence or dieing off of leaves and shoot branches, as sugar is made in the shoot and every bit helps, at least in the beginning of such a crisis.

Scientists have found that there might be a complex starch that acts as a signal for a plant to begin flowering. Say if the root released that, it might travel to the buds in the shoot and start the flowering process.