My husband has psoriatic arthritis and was just worked up for under-active thyroid, so this is all something we've been dealing with lately.
Auto-immune diseases increase the likelihood of under-active thyroid, but it's not always the reason. You should have, if you think it's not working enough (this isn't for over-active) the blood test called Thyroid Antibodies performed as well as TSH and normal thyroid labs.
If the TSH is higher, the thyroid may be under-active. If you have a positive Thyroid Antibody test, it can show your autoimmune disease is attacking your thyroid. Endocrinologists now say to consider treating people outside the range of 0.3 to 3 TSH, which is far narrower than the traditional 0.5 to 5 range they historically followed IF the person is symptomatic.
If you have a TSH of, say, 3.2 and show positive Thyroid Antibodies, you might benefit from a small amount of synthroid, but that would be up to your doctor or, preferably, an endocrinologist to determine for sure.
Some doctors still don't want to look at the narrower range, though, and many would not consider treating someone at 3 or just above, because they're used to letting the TSH go to 5 before they consider it. You just need to first start with a thyroid panel, but as Spooky said, there are other hormone an endocrine issues that can affect things, too, so if thyroid doesn't show anything definitive, you might have to dig deeper.
And just as a side comment, my husband's TSH is high normal in the new range, but his Thyroid Antibodies are sky high (143), and even a well trained endocrinologist didn't want to treat him yet... but suggested he get a thyroid panel q6 months to watch it.