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In mathematics, composite functions and inverse functions are key concepts that expand our understanding of how functions interact with each other. A composite function is formed by combining two functions, where the output of one function becomes the input of another. This process allows for the creation of more complex relationships and is widely used in various branches of mathematics to simplify expressions and solve problems. On the other hand, an inverse function is a function that reverses the effect of the original function. If a function maps an element from one set to another, its inverse will map the second set back to the first. Inverse functions are important for solving equations and understanding the reversibility of operations. In this section, we will examine the definitions, properties, and methods for finding composite and inverse functions, and explore their applications in mathematical analysis and real-world problem-solving.