Composite functions are a fundamental concept in algebra, crucial for understanding calculus and advanced mathematical topics. They involve combining two or more functions to create a new function, where the output of one function becomes the input of another. This skill is frequently tested in competitive exams like JEE.

A composite function is formed by applying one function to the result of another function. If we have two functions, say $f(x)$ and $g(x)$, the composite function $(f \circ g)(x)$ is defined as $f(g(x))$. This means we first evaluate $g(x)$, and then we use that output as the input for $f(x)$.

The composition of functions is denoted by the circle symbol '$\circ$'. So, $(f \circ g)(x)$ means $f(g(x))$. Similarly, $(g \circ f)(x)$ means $g(f(x))$. It's crucial to understand that the domain and range of the composite function are affected by the domains and ranges of the individual functions.

Notice how $(f \circ g)(x) \neq (g \circ f)(x)$ in this example. The order of composition matters significantly.

When finding the composite function $(f \circ g)(x) = f(g(x))$, the domain consists of all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$. The range of $(f \circ g)(x)$ is the set of all values $f(y)$ where $y$ is in the range of $g$ (and $y$ is in the domain of $f$).

Composite functions are building blocks for more complex problems in calculus, such as the chain rule. They appear in questions involving function manipulation, inverse functions, and solving equations that involve nested functions. Mastering this concept is key to unlocking higher-level mathematical skills required for competitive exams.

To excel, practice a variety of problems. Start with simple compositions, then move to those involving rational functions, square roots, and trigonometric functions. Pay close attention to domain and range restrictions. Consider problems where you need to find the composite function itself, evaluate it at a specific point, or work backward to find the original functions.