In mathematics, understanding how to perform operations on functions, such as addition and subtraction, is essential. When adding or subtracting functions, the process mirrors that of combining polynomials. For instance, if we have two polynomials, we can combine like terms to simplify the expression. For example, if we add the polynomials \( f(x) = x^2 + 4 \) and \( g(x) = 5x + 7 \), we can express this as \( f(x) + g(x) = x^2 + 5x + 11 \). This can also be denoted using the notation \( (f + g)(x) \).

When subtracting functions, the same principles apply. For example, if we have \( g(x) - h(x) \), we distribute the negative sign and combine like terms accordingly. It’s crucial to pay attention to the notation, as it can vary, but the underlying operations remain consistent.

Finding the domain of the resulting function after addition or subtraction involves identifying the common values for the individual functions. For instance, if \( f(x) = \frac{1}{x} \) and \( g(x) = x^2 + x + 2 \), the domain of \( f \) excludes \( x = 0 \) due to the denominator, while \( g \) has a domain of all real numbers. Therefore, the domain of \( f + g \) is also restricted to \( x \neq 0 \).

In another example, if we consider \( g(x) = x^2 + x + 2 \) and \( h(x) = x + \sqrt{x - 8} \), we find that the domain of \( h \) is restricted to \( x \geq 8 \) because the expression under the square root must be non-negative. Consequently, the domain of \( g - h \) will also be \( x \geq 8 \), as this is the only restriction present in the operation.

In summary, when adding or subtracting functions, it is important to combine like terms and carefully determine the domain of the resulting function by considering the restrictions imposed by each individual function. This foundational understanding is crucial for further studies in algebra and calculus.