You've asked a tough question! Most people who have studied biology -- myself included -- took the information we were given on faith. It makes sense, it's in a textbook, everyone else seems to believe it... so I will to. There's no experimentation, and there's very little critical thinking. This is how science is taught, but this is not how science is done! We learn much better if we THINK about what we're learning, so it's great that you've asked this question. Many people wouldn't even think to ask it in the first place.

So how did those first scientists studying cell biology discover the role of each organelle? It all starts with observation. Cells weren't even discovered until the microscope allowed us to look at them. It's pretty hard to come up with the idea that cells or organelles exist until you've seen them. Once the microscope became widely available, scientists started to make thin sections of organs and tissues and were able to observe and draw individual cells at high magnification, and compare cell structure from different tissues.

Mitochondria were first discovered in 1886 by Robert Altmann, who thought they were parasites inside the cell. Scientists soon noticed that mitochondria were found in every cell, but were most dense in muscle cells. This suggested that they played an important role in providing the cell with energy.

Special, high-magnification microscopes such as electron microscopes and special techniques for providing very thin slices of tissue allowed scientists not only to see the mitochondria but to see fine structure within the mitochondria. Organelle structure can tell you certain things. For instance, when the inner membranes of an organelle are highly folded, as in mitochondria and chloroplasts, this suggests that an important process is occurring on the membrane surface and increased surface area is important (think of the structure of your lungs, or of fish gills). Also, the outer membrane of mitochondria is very permeable ("open") to molecules, which suggests that the mitochondrial membrane is specialized to transport certain molecules in and out of the cell. (The "permeability" or "openness" of cell membranes is tightly regulated. Can you think why it might be a disadvantage to allow random molecules to enter or leave a cell unregulated?).

Using tiny electrodes inserted into the mitochondria, scientists were able to measure an electrical charge across the inner membrane, which suggested that ions were being actively pumped from one side of the membrane to the other in order to create an "electrochemical gradient". ATP is generated with an enzyme called ATP synthase in the presence of an electrochemical gradient. Scientists were also able to grind up cells and isolate bits of the inner membranes of mitochondria using a device called an ultracentrifuge. When viewed at high magnification, the scientists observed actual ATP synthase enzymes embedded within the membranes. This proved that mitochondria make ATP. ATP is the major source of energy within cells. Bingo!

The role of mitochondria as the energy source ("powerhouse") of the cell was discovered about 50 years ago.

Scientists answer questions about things like mitochondria by doing lots of experiments.

First they had to figure out that everything is made of cells. They started figuring that out after people started inventing the first microscopes, and they could see the cells. Check out a piece of onion skin with a microscope - or maybe the really thin layer that's inside the dry skin. Maybe you'll see cells. I think you can see them better by putting a drop of iodine on the onion skin. You can probably still buy iodine at a pharmacy. And you can buy a good cheap microscope that looks like a little flashlight at places like Radio Shack.

One of the early microscopists looked at a thin slice of cork, which comes from the bark of a tree, and saw the empty cell walls. He thought they looked like the rooms (cells) that monks live in a monastery, so he named them cells.

With better microscopes, scientists saw things inside cells like the nucleus and the mitochondria. They must have named them organelles because they were little things inside cells, like our organs are inside our bodies.

I think that most of cell biology progressed based on the ability to see the organelle but each one has a different set of experiments that led to the discovery of what they do. Still, new jobs for organelles are discovered each year.