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Algebraic functions are mathematical expressions formed using algebraic operations like addition, subtraction, multiplication, division, and taking roots. They typically involve variables and coefficients structured into a polynomial function, rational expressions, or combinations thereof.
In algebraic functions, the variable can be substituted with different values to compute the function's changes, such as seeing how the output changes when the input increases by a small amount (a core aspect of calculus).
To solve problems involving algebraic functions, one needs to substitute the given values into the function and simplify it. This can involve straightforward steps like replacing the variable, or more complex manipulation like expanding terms, factoring polynomials, and solving equations reformulated from the given function.

• Understand the basic structure of algebraic functions (like polynomials).
• Learn to substitute and rearrange terms for simplification.
• Practice expanding expressions and combining quadratic or linear terms.

Polynomial functions are a type of algebraic function consisting of terms of the form \(a_nx^n\) where \(n\) is a non-negative integer, \(x\) is the variable, and \(a_n\) are coefficients. They can be as simple as a constant or as complex as a quintic equation.
The polynomial function in the given exercise is \(f(x) = 3 - 5x + 4x^2\). It is a quadratic polynomial because the highest power of \(x\) is 2. Quadratic polynomials feature prominently in calculations like the difference quotient since they model many natural phenomena.
Understanding polynomial functions involve recognizing standard forms and being able to:

• Identify the degree of the polynomial (the largest exponent of \(x\)).
• Perform arithmetic operations such as addition, subtraction, and multiplication on polynomials.
• Use polynomial identities to simplify expressions, like factoring and expanding.

Substitution is a critical operation in mathematics used to replace variables with numbers or other expressions to simplify and solve problems. It involves directly placing the desired value or expression into every instance of the variable.
In this exercise, substitution is necessary several times:

• First, substituting \(a\) into \(f(x)\) to get \(f(a) = 3 - 5a + 4a^2\).
• Next, substituting \(a+h\) into \(f(x)\) to determine \(f(a+h)\) by replacing each \(x\) in \(f(x)\) with \(a+h\). This requires expanding \((a+h)^2\) and combining like terms.
• Finally, using these subs in the difference quotient formula \(\frac{f(a+h) - f(a)}{h}\).

These steps are common in calculus where the difference quotient is used to determine how functions behave as inputs change minimally, central to grasping derivatives. Mastery of substitution allows for smoother transitions between function evaluation, simplification, and calculus applications.