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Function notation is a way to represent functions using symbols and expressions. It is a mathematical shorthand that makes functions easier to read and manipulate. In function notation, a function is typically expressed as:

• \( f(x) \) where \( f \) represents the function and \( x \) is the input variable.

This notation indicates that the function \( f \) depends on the variable \( x \) and maps it to a specific output.
For example, if we have a function \( f(x) = x - 1 \), it implies that for any input value \( x \), the corresponding output is obtained by subtracting 1 from \( x \).
Function notation helps in organizing and understanding mathematical expressions, especially when dealing with multiple functions like in composition or operations.

Function operations involve combining functions in various ways to create new functions. These operations include addition, subtraction, multiplication, division, and composition.
The composition of functions, denoted as \((fg)(x)\), is an operation where one function \(g\) is applied to the input \(x\), and then the result is used as the input for another function \(f\).

• Mathematically, it is expressed as \((fg)(x) = f(g(x))\).

This means we apply the function \(g\) first and then apply \(f\) on the result of \(g(x)\).
For instance, if \(g(x) = \sqrt{x + 1}\) and \(f(x) = x - 1\), then the composition \((fg)(x) = f(g(x)) = f(\sqrt{x + 1})\), which simplifies to \(\sqrt{x + 1} - 1\).
Function operations are crucial for solving complex problems by breaking them down into manageable steps using simpler functions.

The square root function is a particular kind of function defined as \( g(x) = \sqrt{x} \). It calculates the number that, when multiplied by itself, gives the value of \(x\).

• For example, \( \sqrt{9} = 3 \) because \(3 \times 3 = 9\).
• Similarly, \( \sqrt{16} = 4 \) because \(4 \times 4 = 16\).

The most common version of the square root function includes a constant or another expression inside the square root.
For example, \( g(x) = \sqrt{x + 1} \) is a square root function where 1 is added to \( x \) before taking the square root.
In this context, when finding the composition of functions where one of them is a square root function, we first substitute it accordingly, as shown in the exercise where \( (fg)(x) = \sqrt{x + 1} - 1 \).
Understanding square root functions is essential for dealing with equations involving radicals and helps in solving a variety of practical problems in mathematics and real-life scenarios.