Along with the Free T4 and Free T3, it would also be good if they'd do thyroid antibodies to rule out or confirm Hashimoto's, which is an autoimmune condition that destroys the thyroid
... The antibody tests you need are Thyroid
Peroxidase Antibodies (TPOab) and
Thyroglobulin Antibodies (TgAb). You need them both, since they're both markers for Hashimoto's and some of us have one or the other; some have them both.