My question would be what type of injection?
If the horse was injected with one of the many joint fluid supplements/replacements (most common is Adequan), that's one thing. Many performance horses receive these injections routinely on a preventative basis to prevent joint damage and arthritis. It's your decision whether or not to continue them, however, if the previous owner got good results, I'd be inclined to continue them. How frequently depends on your competition schedule.
If the injections were steroids, or a combination of steroid and Adequan or other joint supplement, that's a big red flag. Steroid injections do little or nothing to help the underlying problem, however, they reduce the inflammation and pain and make the horse temporarily sound. There's nothing wrong with injecting steroids to interrupt the cycle of inflammation and make the horse comfortable, but if a horse needs regular steroid injections to compete, he has a very short competitive career and probably needs a career change.
IMO, steroid injections need to be followed by time off and a change in work routine.
BTW, all of this is my opinion based on my experience and is not to be confused with actual veterinary advice.