Learn Russian: Без этой функции программа работает хуже. - Without this feature, the program works worse.

It acts as an adverb here because it describes the verb работает (works). Хуже is the irregular comparative form for both the adverb плохо (badly/poorly) and the adjective плохой (bad). So here, "works poorly" becomes "works worse" (работает хуже).

Why do we say этой функции instead of эта функция?

The preposition без (without) always forces the words that follow it into the Genitive case. The feminine word эта (this) becomes этой, and the noun функция (feature/function) changes its ending to become функции.

Why does the Genitive form of функция end in -ии instead of just -и or -ы?

Feminine nouns that end in -ия (like функция, история, станция) belong to a special group. In the Genitive case, their ending always becomes -ии.

Does функция always translate to "feature"?

It literally translates to "function", but in the context of software, apps, or technology (like the программа mentioned here), функция is the standard and most natural way to translate the English word "feature".