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A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In this context, we are focusing on single-variable polynomials. A typical polynomial function can be expressed as follows: \( f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \), where each \(a\) represents a constant or coefficient, and \(n\) is a non-negative integer representing degrees.

Consider the polynomial function given: \( f(x) = x^5 \). Here, the function is quite simple as it consists of a single term, known as a monomial, with a degree of 5. The coefficient in this instance is 1, implying the term is simply \(x^5\). Polynomials can represent various types of curves and are vastly used in mathematics for modeling curves and surfaces.

Linear functions are algebraic expressions that form a straight line when graphed. These are among the simplest forms of functions and have the form \( g(x) = mx + b \), where \(m\) is the slope and \(b\) is the y-intercept of the line.

In our case, we have \( g(x) = 7x - 1 \). This function is linear because it represents a straight line where every increase in \(x\) by 1 results in an increase in \(g(x)\) by 7, and it intersects the y-axis at \(b = -1\). Linear functions are fundamental in mathematics and help to establish the basics of more complex functions.

Substitution is a core concept in function operations, especially when dealing with composite functions. When substituting, you replace the variable in a function with another function or expression. The goal is to evaluate a function in terms of another.

For example, to find \(f(g(x))\), substitute \(g(x)\) into every \(x\) in \(f(x)\). If \(f(x) = x^5\) and \(g(x) = 7x - 1\), then \(f(g(x)) = (7x - 1)^5\). Similarly, to find \(g(f(x))\), replace \(x\) in \(g(x)\) with \(f(x)\), making \(g(f(x)) = 7(x^5) - 1 = 7x^5 - 1\). Substitution enables transformations of functions through other mappings.

Function composition involves combining two functions in such a way that the output of one function becomes the input of another. Notated as \((f \circ g)(x)\), it represents \(f(g(x))\) and is computed by following a sequential process of substitution, essentially substituting one function into another.

When calculating compositions like \(f(g(x))\), the function \(g(x)\) is computed first, and this result is used as the input for the function \(f(x)\). Therefore, with our example where \(f(x) = x^5\) and \(g(x) = 7x - 1\), we compute \(f(g(x)) = (7x - 1)^5\). Likewise, for \(g(f(x))\), the roles reverse, and \(f(x)\) serves as the input for \(g(x)\), resulting in \(7x^5 - 1\). Calculating function compositions is a crucial operation in mathematics, enabling complex operations and models derived from simpler functions.