Question

Use substitution to compose the two functions. $$ w=5 s^{3} \text { and } q=3+2 w $$

Short answer

Answer: The function \(q\) in terms of \(s\) is \(q=3+10s^3\).

Step-by-step solution

Understand the functions and the substitution

We are given two functions, \(w=5s^3\) and \(q=3+2w\). Our goal is to substitute the function \(w\) in the function \(q\) which will give us a new function in terms of \(s\).

Substitute the w function into the q function

Now, we need to substitute the function \(w=5s^3\) into the function \(q=3+2w\). To do this, we replace the \(w\) in the \(q\) function with \((5s^3)\). Thus, $$q=3+2(5s^3)$$

Simplify the composed function

Now, let's simplify the composed function by distributing the \(2\) to the terms inside the parenthesis: $$q=3+10s^3$$ Now, we have successfully composed the two functions and have found the new function \(q\) in terms of \(s\).

Key concepts

Substitution Method

The substitution method is a powerful technique used in mathematics to replace variables in one expression with their equivalent from another expression. This method is often used in function composition to create a new function from existing ones. The goal is to simplify or transform complex equations by expressing them in different forms that are easier to work with.

To effectively use the substitution method:

• Identify the functions involved and the variable to be substituted.
• Clearly define what needs to be replaced and what it will be replaced with.
• Ensure that during substitution, the expression maintains its original mathematical properties and meanings.

In our exercise, we applied the substitution method by replacing the variable \(w\) in the function \(q = 3 + 2w\) with the expression \(5s^3\) from the function \(w = 5s^3\). This action forms a new and simplified expression in terms of \(s\), allowing us to handle the equation more easily in further calculations or analysis.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operation symbols. They represent quantities in a generalized form, allowing for flexible manipulation and analysis.

In the context of our exercise, the algebraic expressions involved are:

• \(w = 5s^3\), where \(w\) is expressed as a function of \(s\).
• \(q = 3 + 2w\), where \(q\) depends on \(w\).
• After substitution, we get \(q = 3 + 10s^3\), an expression that relates \(q\) directly with \(s\).

These expressions allow us to work with abstract numbers and relationships in a controlled manner, helping us solve equations and model real-world problems.

Understanding how to work with algebraic expressions is essential, as they form the building blocks of algebra and various levels of mathematics.

Simplification of Expressions

Simplification of expressions involves reducing an expression to its most basic form while maintaining its original value. This process often involves combining like terms and performing arithmetic operations within the expression.

In our example, once the substitution is done, the expression \(q = 3 + 2(5s^3)\) needs simplification:

• Multiply the term inside the parenthesis by 2, giving \(10s^3\).
• Then, add this result to 3, resulting in \(q = 3 + 10s^3\).

Simplification helps in making expressions clearer and more manageable, often leading to easier problem-solving and a better understanding of the mathematical relationships involved.

By simplifying expressions, we not only make equations tidier but also prepare them for further mathematical operations, such as differentiation or integration if required.