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Algebraic functions are mathematical expressions involving numbers and variables involving operations such as addition, subtraction, multiplication, division, and exponentiation to fixed powers. A function is defined as a relationship where each input has a single output.
In the given problem, we deal with two specific algebraic functions:

• Function \( f(x) = 3x + 5 \)
• Function \( g(x) = 5 - x \)

These functions are linear because they involve linear expressions where the variable \( x \) is raised to the power of 1. The coefficients and constants determine the shape and position of their graph on the Cartesian plane, making algebraic functions essential in representing various real-world problems, such as calculating cost, speed, and growth.

Composite functions involve the combination of two functions where the output of one function becomes the input of another. This is denoted by the symbol \( \circ \), which stands for composition.
When executing composition, such as \( (f \circ g)(x) = f(g(x)) \), we replace \( x \) in the function \( f(x) \) with the entire function \( g(x) \). For our example:

• \( (f \circ g)(x) = f(g(x)) = 3(5 - x) + 5 = 20 - 3x \)
• \( (g \circ f)(x) = g(f(x)) = 5 - (3x + 5) = -3x \)
• \( (f \circ f)(x) = f(f(x)) = 3(3x + 5) + 5 = 9x + 20 \)

Composite functions provide a tool for linking different processes together and are common in sequences of operations in more complex mathematical problems.

Function operations refer to various ways to manipulate and interact functions, such as addition, subtraction, multiplication, division, and composition. These operations help to understand how functions behave and interact with each other. Composition is one specific type of function operation, but other operations can be equally useful.
To perform composition, it's crucial to correctly substitute the functions and pay attention to the order of operations:

• Simplify by carefully substituting the entire expression of one function into another
• Maintain accurate arithmetic throughout the simplification

Understanding how to handle function operations ensures that complex equations are simplified correctly, making it easier to solve various algebraic problems. Practicing different types of operations expands problem-solving skills and deepens comprehension of how different mathematical concepts are connected.