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Function addition involves combining two functions by adding their corresponding outputs. In mathematical terms, if you have two functions, say, \(f(x)\) and \(g(x)\), the addition of these functions can be written as \((f+g)(x) = f(x) + g(x)\).
This means that for every input value \(x\), you get the output by adding the individual outputs of \(f\) and \(g\).
For our specific example, we have \(g(x) = 2x\) and \(h(x) = x - 3\). To add these functions:

• First, substitute the individual functions: \((g+h)(x) = 2x + (x - 3)\).
• Next, combine like terms to simplify: \(2x + x - 3 = 3x - 3\).

Function addition is commonly used to simplify and combine multiple functions into one. By performing function addition, we turn two distinct functions into a single, more manageable function, which can then be evaluated or used for further operations.

Combined functions (also known as composite functions) are created by performing operations on two or more functions. This can be addition, subtraction, multiplication, or division. In our case, we are dealing with the addition of functions.
By combining functions, we aim to merge them into one operational unit. This is particularly useful when functions are related and we want to see their cumulative effect.
In the problem at hand, we combine \(g(x)\) and \(h(x)\):

• First, identify the individual functions: \(g(x) = 2x\) and \(h(x) = x - 3\).
• Then, perform the addition: \((g+h)(x) = 2x + (x - 3)\), which simplifies to \(3x - 3\).

Combining functions allows us to take two separate mathematical expressions and make a new function that can be evaluated or used for various calculations.
This technique is immensely helpful in solving complex problems by breaking them down into simpler additions or other basic operations.

Function evaluation is the process of calculating the output of a function for a specific input value.
To evaluate a function, you substitute the input value into the function and perform the necessary calculations.
In the given exercise, we need to evaluate the combined function \((g+h)(x)\) at \(x = 1\):

• First, we have the combined function: \((g+h)(x) = 3x - 3\).
• Substitute \(x = 1\): \((g+h)(1) = 3(1) - 3\).
• Simplify the expression: \(3 - 3 = 0\).

The result is \((g+h)(1) = 0\).
Function evaluation helps us determine specific values of the function, which is crucial in many practical scenarios, such as solving equations or graphing functions.
Through function evaluation, we get concrete numbers that represent the behavior of the function at particular points.