Is the discovery of Alien or Extra Terrestrial Life a function of the science of Astronomy? When i read astronomy articles on the search for new Planets or discussions on Mars a lot of the topic is on the planets occurring in the habitable zone and the possibility of life on them or stories on the possibility of previous life on Mars. Is this actually part of the science objectives for the Astronomy community, or is there another branch of science who is tasked with this? Is Astronomy only focused on the mechanics of celestial bodies and the discovery of Alien or Extra Terrestrial life is just and interesting sidebar?
Certainly it's a function of astronomy - chances are, if you're doing science outside the Earth then by definition it's astronomy in some form or another. There are many sub-fields of astronomy that are useful in the search for extra-terrestrial life:
All of these sub-fields contribute very interesting and valuable information towards what is by no means a simple question to answer.
The search for extraterrestrial life is usually covered by "exobiology". But astronomy can provide clues of whether a planet or an exoplanet provides some of the conditions, which allow life or indicate life.
The habitable zone - in its various versions - defines, in which distance from a star a planet needs to be to allow for e.g. liquid surface water, a condition regarded as necessary for most life-forms we know from Earth. With the discovery of extremophiles the habitable zone for that kind of life has been extended.
Astronomy can provide information about the central star and the distance of a planet from the central star. From these data a decision is possible, whether the planet fulfills the surface temperature condition for habitability.
Astronomy can provide information about the stability of a star, necessary for life to evolve on its planet.
Astronomy can - in principle - provide information about the chemical composition of a potential atmosphere of a planet. Oxygen and methane at the same time would be regarded as an indicator of possible life on the planet.
Astronomy can determine size and density information about the planet. That way it can be decided, whether it's a rocky planet or a gas planet, and infer the gravitational acceleration at the surface. The gravitational acceleration is relevant, e.g. for constraints of the atmospheric composition, or for possible properties of life, if extant.