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In mathematics, the domain of a function is a set of all possible input values (commonly known as 'x-values') that will not lead to any undefined expression within the function. These values allow us to perform the necessary calculations without breaking the rules of mathematics. For instance, when considering a function that involves a square root, the expression under the square root must be non-negative to remain within real numbers. This is because the square root of a negative number does not produce a real number. When evaluating a domain, consider these key points:

• Ensure there are no divisions by zero, which would make the function undefined.
• Expressions inside square roots must be non-negative, meaning they must be greater than or equal to zero.
• For functions with logarithms, the argument must be a positive number.

In the original problem, we worked with the function \( g(x) = \sqrt{x - 1} \). Therefore, to find the domain, we solve \( x - 1 \geq 0 \), leading us to \( x \geq 1 \). This means for any \( x \) value less than 1, \( g(x) \) is undefined.

The square root function, often expressed as \( g(x) = \sqrt{x} \), is a specific type of function involving the mathematical operation of taking a square root. This function arises frequently in different areas of mathematics and has some specific properties. Notably, the square root function is only defined for non-negative values. Understanding how to evaluate square root functions is critical:

• Confirm the input value makes the expression under the square root non-negative.
• If the input alters the function to include negative results under the square root, those inputs are not in the domain.
• The output of a square root function represents a non-negative number.

In our exercise example, we calculated \( g(2) = \sqrt{2-1} = \sqrt{1} = 1 \). We verified that \( 2 \) is within the domain of \( g(x) \), because \( 2 - 1 = 1 \geq 0 \). Thus, it produces a valid and defined result.

Combined functions refer to the operation where two or more functions are combined together, either through addition, subtraction, multiplication, or division. These operations create a new function which behaves in a predictable way based on the individual functions. When combining functions, one of the most common operations is addition, referred to as \( (f+g)(x) \). Evaluating combined functions involves:

• First, individually evaluating each function at the specified input value.
• Ensuring all values used are within the domain of each specific function involved in the combination.
• Adding the results of the individual evaluations.

In the exercise provided, first \( f(2) = 14 \) and \( g(2) = 1 \) were computed. Then, by adding these results together, the evaluation of \( (f+g)(2) \) is \( 15 \). This illustrates how to handle combined functions and shows the importance of checking each function's domain before performing operations.