Finite math and precalculus both refer to math before calculus. Finite math, however, is a catch-all title representing any math before calculus, while precalculus is more narrowly defined as the algebra knowledge necessary to perform calculus, often called algebra 3. If you intend to move on to calculus and beyond, precalculus is highly recommended, if not necessary, over finite math due to the difference in algebra skills gained during the course.
TL;DR (Too Long; Didn't Read)
If you're going to study calculus, a precalculus course is immensely helpful in preparing you to make some big conceptual leaps. If you're not going on to calculus a finite math course might be more immediately useful, depending on your career plans.
If you understand calculus, then understanding finite math and precalculus is easier, as the latter two are simply what calculus is not. Calculus is the next advanced class after algebra and precalculus, and it introduces students to the great conceptual leaps of differentiation and integration. Differentiation allows you to take apart mathematical functions to understand their behavior, while integration lets you put them back together, adding together small numbers. You must have strong algebra skills to be successful in calculus.
The Goal of Finite Math
In finite math classes, the goal is to give students enough information to use mathematical analysis in the real world, at jobs or at home. Topics covered include matrix algebra, probability, statistics, logic and discrete mathematics. You learn simple, immediately useful ways to count, calculate, add, subtract, multiply and divide. While success in finite math can be immensely helpful in the real world, it does not necessarily prepare you for a full calculus class.
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The Value of Precalculus
Precalculus, also called algebra 3, is the highest-level algebra class you can take before going into calculus. In this course, you become comfortable with quantitative literacy and logic, such as algorithms, logic and proofs, functions, geometry, trigonometry, statistics and probability. You learn how to form relationships between numbers in a way that provides you more information about what the numbers mean. For example, that might mean solving for an unknown variable by constructing an equation. You also become more comfortable manipulation those variables that stand in for unknown numbers.
The differences between finite math and precalculus are nuanced, often hidden in the details of the two courses. You will gain a wider variety of mathematical knowledge in finite math, but not all of this knowledge is useful in calculus. In precalculus, everything taught is done with the intention that it will help you in calculus. When you then go on to take a calculus course, you'll see how necessary that background in algebra and critical thinking has become. In finite math, and even precalculus, some students can memorize patterns and pass the class. However, due to the nature of calculus and the level of integrated thinking between concepts, you must demonstrate a deeper understanding of the theory behind the math to be successful in a calculus class.