A 45-year-old woman has lost 12 kg and experiences episodes of rapid heartbeat. What is the most appropriate investigation to perform?

The most appropriate investigation in this scenario is the thyroid function test. The patient's significant weight loss and episodes of rapid heartbeat suggest hyperthyroidism as a potential underlying cause. Hyperthyroidism is characterized by increased levels of thyroid hormones, which can lead to symptoms such as weight loss, increased heart rate, anxiety, and heat intolerance.

Thyroid function tests typically include measurements of serum levels of Thyroid Stimulating Hormone (TSH) and thyroid hormones, namely T3 and T4. In cases of hyperthyroidism, TSH levels are usually suppressed, while T3 and T4 levels are elevated. This test allows for a direct assessment of the thyroid's activity and is crucial for diagnosing thyroid disorders.

Other investigations, such as thyroid antibodies, are more specific for certain autoimmune thyroid conditions like Graves’ disease but are not immediately necessary to confirm hyperthyroidism. An adrenocorticotropic hormone stimulation test is used for evaluating conditions related to adrenal function and is not relevant in this case. An echocardiogram can assess heart structure and function but does not address the underlying issue of thyroid status related to the symptoms presented. Thus, the thyroid function test is the most appropriate initial investigation in this scenario.