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Function addition is a process where we combine two functions by adding their corresponding outputs. If you have two functions, \(f(x)\) and \(g(x)\), the sum of these functions, denoted as \((f + g)(x)\), is calculated by adding the values of \(f(x)\) and \(g(x)\) at each \(x\).

For example, if we need to find \((f + g)(2)\), we first look up the values of \(f(2)\) and \(g(2)\) from the given tables. From our tables, \(f(2) = 5\) and \(g(2) = -11\). We simply add these values together to obtain:

\ \[(f + g)(2) = f(2) + g(2) = 5 + (-11) = -6\]

Using this method, you can easily calculate sums for any other value of \(x\). Function addition is useful for combining different datasets or for simplifying complex functions by breaking them into simpler components.

Function subtraction works similarly to function addition, but instead of adding the corresponding outputs, we subtract them. For two functions, \(f(x)\) and \(g(x)\), the result of their difference, denoted as \((g - f)(x)\), is given by subtracting the value of \(f(x)\) from \(g(x)\).

For instance, if we need to find \((g - f)(0)\), we check the tables for \(g(0)\) and \(f(0)\), which are both \(-3\). Thus:

\ \[(g - f)(0) = g(0) - f(0) = -3 - (-3) = 0\]

Subtracting functions helps highlight differences between two datasets or to simplify a problem by isolating specific effects. Practice more with other values to become confident in this essential skill.

Function multiplication involves multiplying the outputs of two functions at each point. Given two functions, \(f(x)\) and \(g(x)\), their product is represented as \((f \cdot g)(x)\). The values of the functions are simply multiplied together for each \(x\).

For instance, if you need to find \((f \cdot g)(3)\), locate the values \(f(3)\) and \(g(3)\) in the tables: \(f(3) = 15\) and \(g(3) = -21\). Now multiply them:

\ \[(f \cdot g)(3) = f(3) \cdot g(3) = 15 \cdot (-21) = -315\]

This method is useful for combined effects or determining how two different variables interact.

Function division is the process of dividing the outputs of two functions. For two functions, \(f(x)\) and \(g(x)\), the quotient is represented as \(\left(\frac{g}{f}\right)(x)\). This operation involves dividing the value of \(g(x)\) by the value of \(f(x)\) for each \(x\).

As an example, to find \(\left(\frac{g}{f}\right)(4)\), we use the tables to find \(g(4)\) and \(f(4)\): \(g(4) = -35\) and \(f(4) = 29\). The calculation is then:

\ \[\left(\frac{g}{f}\right)(4) = \frac{g(4)}{f(4)} = \frac{-35}{29}\]

Function division is particularly useful for understanding ratios and relationships between datasets. Be cautious, however, and ensure \(f(x) eq 0\) to avoid division by zero.