Functions are a fundamental concept in mathematics. A function is a rule that assigns each input value from a set called the domain to a unique output value from a set called the range. It is often denoted as f(x), where x represents the input value. For example, f(x) = 2x + 3 is a function that takes an input x, multiplies it by 2, and then adds 3 to it. Let's understand some key properties of functions:

1. Domain: The domain of a function is the set of all possible input values. It determines the valid values for x in the function. For example, if we have a function f(x) = sqrt(x), the domain is all non-negative real numbers, as the square root of a negative number is undefined.

2. Range: The range of a function is the set of all possible output values. It represents the values that the function can produce. Using the same example, the range of f(x) = sqrt(x) is all non-negative real numbers, as the square root of a non-negative number is always non-negative.

3. Inverse Functions: An inverse function undoes the action of the original function. If a function f(x) outputs y, the inverse function, denoted as f^(-1)(y), takes y as input and returns the original input x. To determine if two functions are inverses, we must ensure that their compositions yield the identity function, f(f^(-1)(y)) = y.

Understanding these properties will provide a solid foundation for exploring more advanced topics in functions.

Remember, practice makes perfect! Try solving some sample questions and exercises to reinforce your understanding of functions and their properties. Keep up the great work!