This pattern turns a yes/no question into a noun-like clause that can be the subject or object of a sentence. It is very common in everyday speech and writing. For example, you can say 行くかどうか (whether or not I go) or 高いかどうか (whether or not it's expensive). Note that for na-adjectives and nouns, you include だ before か: 元気かどうか, 日本人かどうか.

You can also use it with past tense: 行ったかどうか, 高かったかどうか. The clause can be followed by verbs like 知らない (don't know), 聞く (ask), 決める (decide), or 分かる (understand). It often answers an implicit question: 'Do you know whether he will come?' = 彼が来るかどうか知っていますか.

In casual conversation, you might hear just か (without どうか) in some contexts, but かどうか is standard and safer for learners. It is not overly formal and works in both speech and writing.