In the US you are not required to have a corporation to use business expenses to offset your income. The technical term you need is "deducting business expenses", and in matters of taxes it's usually best to go straight to the horse's mouth: the IRS's explanations
Deducting Business Expenses
Business expenses are the cost of carrying
on a trade or business. These expenses are usually deductible if the
business operates to make a profit.
What Can I Deduct? Cost of Goods
Sold, Capital Expenses, Personal versus Business Expenses, Business Use
of Your Home, Business Use of Your Car, Other Types of Business Expenses
None of this requires any special incorporation or tax arrangements, and are a normal part of operating a business.
However, there is a bit of a problem with your scenario. You said you "invested" into a business, but you mentioned buying specific things for the business which is not generally how one accounts for investment. If you are not an owner/operator of the business, then the scenario is not so straight-forward, as you can't simply claim someone else's business expenses as your own because you invested in it.
Investments are taxed differently than expenses, and based upon your word choices I'm concerned that you could be getting yourself into a bit of a pickle. I would strongly advise you to speak with a professional, such as a Certified Public Accountant (CPA), to go over your current arrangement and advise you on how you should be structuring your ongoing investment into this shared business.
If you are investing you should be receiving equity to reflect your ownership (or stock in the company, etc), and investments of this sort generally cannot be deducted as an expense on your taxes - it's just an investment, the same as buying stock or CDs. If you are just buying things for someone else's benefit, it's possible that this could be looked upon as a personal gift, and you may be in a precarious legal position as well (where the money is, indeed, just a gift). And gifts of this sort aren't deductible, either.
Depending on how this is all structured, it's possible that you should both consider a different form of legal organization, such as a formal corporation or at least an official business partnership. A CPA and an appropriate business attorney should be able to advise you for a nominal (few hundred dollars, at most) fee. If a new legal structure is advisable, you can potentially do the work yourself for a few hundred dollars, or pay to have it done (especially if the situation is more complex) for a few hundred to a few thousand. That's a lot less than you'd be on the hook for if this business is being accounted for improperly, or if either of your tax returns are being reported improperly!