Can a purchased app for IPHONE be used for my IPAD?

Transfer them from your iPhone to iTunes and then sync them to your iPad.

Or if away from computer, just re-download it again as long as you are logged in with the same iTunes account. You will get a pop-up saying that you already purchased the item, would you like to download again for free. So, 2 ways to do it.

iPhone/Touch apps will be played at the same size, centered on the iPad. Little 2x button in the corner will allow you to play it twice the size, although it will be more pixelated. Some apps still look good that way though.

Checking my Apps sections under Library I have:

iPhone, iPod touch, and iPad Apps
This section has your Universal apps. Where one download works on the iPhone or iPad.

iPhone and iPod touch Apps
These are only iPhone apps. They will run in a small window, or can be ran as 2x size on the iPad.

iPad Apps
These are only iPad apps and cannot be installed on the iPhone.

To sync them up, when you plug in your iPad, you should see it under Devices. Click on it.

Your iPad Summary tab should come up. Click on Apps tab.

On the Sync Apps checkbox, make sure it is checked. Then below you will see your list of apps on the left, and your iPad screens on the right. Here can can check on or off individual apps, and move them around your iPad screens, etc.

When done, click Apply.