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An algebraic function is a type of function that can be expressed using polynomial equations with a finite number of terms, and where each term is made of constants, variables, and exponents. These functions form the basis of algebra and are crucial for understanding various concepts within mathematics.

The key characteristic of an algebraic function, like the one in our exercise, is that it involves the composition of variables raised to whole number powers and constants. In this case, the algebraic function given is \( f(x) = \frac{1}{x^{3} + 1} \). It includes a polynomial in the denominator and the constant 1 in the numerator, which is a good example of a simple rational function, a common type of algebraic function.

The function can be used to map every input value of \( x \) to exactly one output value. When you are asked to find the value of the function for a specific input, like \( f(2) \) or \( f(y^2) \), you're essentially finding what value the function takes for that particular input.

The substitution method is a fundamental technique in algebra used to evaluate expressions, including functions. It involves replacing the variables in an expression with given numbers or other expressions. This process is crucial when we need to calculate the value of function for a particular input.

To apply the substitution method to our exercise, you take the original function \( f(x) = \frac{1}{x^{3} + 1} \) and substitute the variable \( x \) with the value you're evaluating the function at. For instance, with \( f(2) \), we replace every occurrence of \( x \) with 2, and with \( f(y^2) \), we substitute \( x \) with \( y^2 \).

Example of Substitution

For \( f(2) \), after substitution we get \( \frac{1}{2^{3} + 1} \), and for \( f(y^2) \), the substitution gives us \( \frac{1}{(y^2)^{3} + 1} \). It's like putting the value into the original formula's place and then continuing with the simplification.

Simplifying expressions is the process of making an algebraic expression easier to understand and to work with by performing operations and combining like terms. This process often includes arithmetic operations such as addition, subtraction, multiplication, division, and exponentiation, and follows the order of operations rules.

In our exercise, after substituting the values into the function, we simplify the expressions to find the result. The process might involve arithmetic operations to combine like terms or to reduce the expression to its simplest form. For example, simplifying the expression \( f(2) \) involves calculating the value of \( 2^{3} \) and then adding 1 to get 9, allowing us to simplify it further to \( \frac{1}{9} \).

Remember, the goal of simplification is to rewrite expressions in a way that makes them more understandable without changing their value. This principle is important not just for evaluating specific cases, like in this exercise, but also in solving equations and understanding the general behavior of algebraic functions.