Daring Discovery

If you can't cast the discovered card (perhaps because there are no legal targets for the spell), you'll put it into your hand.

You exile the cards face up. All players will be able to see them.

If you cast a spell "without paying its mana cost", you can't choose to cast it for any alternative costs. You can, however, pay additional costs. If the spell has any mandatory additional costs, you must pay those to cast it.

You can cast Daring Discovery with no targets and just discover 4. However, if you choose any targets, and all of those targets are illegal by the time Daring Discovery tries to resolve, it won't resolve and none of its effects will happen. You won't discover 4. As long as one target remains legal, any legal targets won't be able to block this turn, any illegal targets won't be affected, and you'll discover 4.

A spell's mana value is determined only by its mana cost. Ignore any alternative costs, additional costs, cost increases, or cost reductions.

The mana value of a split card is determined by the combined mana cost of its two halves. If discover allows you to cast a split card, you may cast either half (as long as its mana value is less than or equal to the effect's discover value) but not both halves.