It's a "reasonable start", even if the language is a bit old-fashioned. There's a bit of "overlap" with what is commonly called "Pre-Calc" these days, but it's more of a thorough exploration of algebra, then it is of functions that the calculus is typically concerned with. So there's still a bit of a "gap" there, but after reading this book, you should be in good shape to start to cross it.
Some advice, not just targeted at this book: reading math is a bit like drinking wine, gulping it down doesn't do it justice. Read it once, read it again just to be sure, and try to work out every problem you can (you can, if you like, skip the ones you can solve in your head, as long as you feel that you COULD solve them, in front of witnesses on a timer, with your life at stake).
Another book I can heartily recommend is THIS one:
Algebra: Israel M. Gelfand, Alexander Shen: 9780817636777: Amazon.com: Books
which is an elementary algebra textbook written by a first-class mathematican. It's rather small size belies the depth of information it covers (as one reviewer gushed: "it closes with a proof of the Cauchy-Schwartz inequality!"). It is much more "modern" in tone.