Question

Use substitution to compose the two functions. $$ P=3 q^{2}+1 \text { and } q=2 r^{3} $$

Short answer

Question: Determine P as a function of r given that P is a function of q and q is a function of r, where P(q) = 3q^2 + 1 and q(r) = 2r^3. Answer: P'(r) = 12r^6 + 1

Step-by-step solution

Identifying the given functions

First, let's write down the two given functions: $$ P(q) = 3q^2 + 1 \text{ and } q(r) = 2r^3 $$

Composing the functions

Now, we must substitute the expression for q(r) into the function for P(q). Let the composed function be defined as P'(r), for simplicity: $$ P'(r) = P(q(r)) = 3(2r^3)^2 + 1 $$

Simplifying the composed function

Next, we need to simplify the expression for P'(r). First, square the term inside the parentheses: $$ P'(r) = 3(4r^6) + 1 $$ Then, multiply 3 by the term inside the parentheses: $$ P'(r) = 12r^6 + 1 $$

Writing the final answer

The composed function P'(r) is: $$ P'(r) = 12r^6 + 1 $$

Key concepts

Substitution Method

The substitution method is a powerful technique used in mathematics to simplify complex expressions by replacing one variable with an expression derived from another function. This approach allows us to find connections between different functions and simplify our calculations.
For example, in the given exercise, you start with two functions: \( P(q) = 3q^2 + 1 \) and \( q(r) = 2r^3 \). To find the composed function, you substitute the expression for \( q(r) \) into the function for \( P(q) \). By replacing \( q \) in \( P(q) \) with \( 2r^3 \), we form a new function \( P'(r) = P(q(r)) \).
This substitution transforms the problem by allowing us to handle just one variable at a time, streamlining the evaluation process.

Polynomial Functions

Polynomial functions are expressions composed of variables and constants using addition, subtraction, multiplication, and non-negative integer exponents. These functions play a central role in algebra and calculus due to their straightforward properties and wide range of applications.
The exercise involves the polynomial function \( P(q) = 3q^2 + 1 \). Here, \( q^2 \) signifies a squared term, and each component follows the basic rules of polynomial arithmetic.
Analyzing polynomial behavior:

• The degree of the polynomial: The highest exponent in the term, which in this case is 2 (from \( q^2 \)).
• Coefficients: The numbers in front of the terms, such as 3 for \( q^2 \) and the constant term is 1.
• Shape of the graph: Degree 2 polynomials, like this one, form parabolas when graphed.

Understanding these features can help in solving many algebraic problems with ease.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra, which helps in reducing complex mathematical expressions into more manageable forms. This involves combining like terms, using exponents laws, and performing arithmetic operations efficiently.
In our exercise, simplifying the expression \( P'(r) = 3(2r^3)^2 + 1 \) is critical to obtaining the final composed function.
Steps to simplify:

• Square the function inside the parentheses \((2r^3)^2\), which results in \(4r^6\) because you square both the constant and the variable.
• Multiply the coefficient: Multiply 3 by the simplified term \(4r^6\) to get \(12r^6\).
• Add the constant term: Finally, add 1 to complete the simplification, yielding \(12r^6 + 1\).

Simplifying expressions accurately allows for clearer comprehension and easier problem-solving in mathematical operations.