Certainly! We are given two functions:
1. [tex]\( P(q) = 7q^2 + 9 \)[/tex]
2. [tex]\( q(r) = 3r^3 \)[/tex]
We want to compose these two functions by substituting [tex]\( q(r) \)[/tex] into [tex]\( P(q) \)[/tex].
Step-by-step, here’s how we do this:
1. Start with the definition of [tex]\( P(q) \)[/tex]:
[tex]\[ P(q) = 7q^2 + 9 \][/tex]
2. Substitute [tex]\( q(r) = 3r^3 \)[/tex] into [tex]\( P(q) \)[/tex]. This means that everywhere there is a [tex]\( q \)[/tex] in the function [tex]\( P \)[/tex], we will replace it with [tex]\( 3r^3 \)[/tex]:
[tex]\[ P(r) = 7(3r^3)^2 + 9 \][/tex]
3. Now, simplify the expression inside the parentheses:
[tex]\[ (3r^3)^2 \][/tex]
This means raising [tex]\( 3r^3 \)[/tex] to the power of 2, which results in:
[tex]\[ (3r^3)^2 = 3^2 \cdot (r^3)^2 = 9r^6 \][/tex]
4. Substitute this back into the function for [tex]\( P \)[/tex]:
[tex]\[ P(r) = 7 \cdot 9r^6 + 9 \][/tex]
5. Simplify the constants by multiplying:
[tex]\[ P(r) = 63r^6 + 9 \][/tex]
Therefore, after performing the substitution and simplification, the composed function [tex]\( P \)[/tex] in terms of [tex]\( r \)[/tex] is:
[tex]\[ P = 63r^6 + 9 \][/tex]
