The expansion of the early universe, known as cosmic inflation, did indeed occur at a rate faster than the speed of light. However, this expansion is not responsible for making the universe appear younger than it actually is.
The concept you're referring to is often called the "horizon problem" in cosmology. It arises from the observation that different regions of the universe, which are now too far apart to have had any direct contact, appear to have the same properties and temperature. This seems puzzling because light from one region would not have had enough time to reach the other and equalize their temperatures.
The solution to the horizon problem lies in the theory of cosmic inflation. During the inflationary period, the universe expanded exponentially, stretching regions that were in close proximity before inflation to be much larger than our current observable universe. This rapid expansion homogenized the early universe, ensuring that distant regions that are now separated from each other were once in close contact.
After the inflationary phase ended, the universe continued to expand, but at a slower rate. Light from distant galaxies could then travel through space and reach us over the billions of years that followed. So, the light from early galaxies wasn't "carried along" by the rapid expansion but rather had enough time to reach us after inflation when the expansion was slower.
The expansion of the universe doesn't affect the age we assign to it based on the light we receive from distant galaxies. Astronomers determine the age of the universe using various methods, including the measurement of the cosmic microwave background radiation and the observations of the Hubble constant, which relates the rate of expansion to the distance of galaxies.
In summary, while the expansion of the early universe occurred faster than the speed of light during cosmic inflation, it didn't make the universe appear younger than it actually is. The light from early galaxies had enough time to reach us after inflation, and the age of the universe is determined through other observational methods.