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The substitution method is a useful technique in function composition where one function is substituted into another. This method allows us to create a new function by plugging one function into the variable of another. It requires understanding the input and output relationship between the functions.
For example, consider the functions given: \(w(r) = r^2 + 5\) and \(r(t) = t^3\). The goal is to substitute \(r(t)\) into \(w(r)\).
We do this by replacing every occurrence of \(r\) in the function \(w(r)\) with the function \(r(t)\). So,\(w(t) = (t^3)^2 + 5\). This new expression represents the composed function.
Remember, it's important to correctly identify which parts of the function need to be substituted and to verify that substitution leads to the correct variable being expressed in terms of the desired input.

Polynomial functions are expressions consisting of variables raised to whole number powers, combined using addition, subtraction, and multiplication. They play a key role in algebra and calculus, appearing in diverse areas of mathematics and engineering.
In the exercise, both functions \(w(r) = r^2 + 5\) and \(r(t) = t^3\) are polynomial functions:

• \(w(r)\) is a quadratic polynomial because the highest power of \(r\) is 2.
• \(r(t)\) is a cubic polynomial because \(t\) is raised to the power of 3.

When we compose these functions, \(w(t) = t^6 + 5\), the result is a polynomial of degree 6. The composition combines the powers of the variables, demonstrating why polynomial functions are structured as they are, often making them straightforward to manipulate and analyze.

Order of operations is an essential principle ensuring that mathematical expressions are solved uniformly and correctly. It is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) in some regions.
Applying this rule guarantees that calculations are performed in the correct sequence, preventing errors.
In our composed function \(w(t) = (t^3)^2 + 5\), following the order of operations is crucial:

• First, we handle exponents: \((t^3)^2\) becomes \(t^6\).
• Next, we perform addition: adding 5 to \(t^6\).

Thus, the correctly simplified function is \(w(t) = t^6 + 5\), ensuring that we have followed the appropriate steps for accurate results. Mastery of order of operations helps in resolving complex problems and prevents common errors in arithmetic calculations.