To compose the two functions [tex]\( p \)[/tex] and [tex]\( D \)[/tex] using substitution, follow these steps:
1. Define the Function [tex]\( p \)[/tex]:
The function [tex]\( p \)[/tex] is given by:
[tex]\[
p(q) = 9 q^9
\][/tex]
2. Define the Function [tex]\( D \)[/tex]:
The function [tex]\( D \)[/tex] is given by:
[tex]\[
D(p) = 8p - 3
\][/tex]
3. Substitute [tex]\( p(q) \)[/tex] into [tex]\( D(p) \)[/tex]:
We need to express [tex]\( D \)[/tex] in terms of [tex]\( q \)[/tex] by substituting the expression for [tex]\( p \)[/tex] into the function [tex]\( D \)[/tex].
Start with the function [tex]\( D \)[/tex]:
[tex]\[
D(p) = 8p - 3
\][/tex]
Substitute [tex]\( p(q) = 9 q^9 \)[/tex] into this equation:
[tex]\[
D(p(q)) = 8(9 q^9) - 3
\][/tex]
Simplify the expression inside the parentheses:
[tex]\[
D(p(q)) = 72 q^9 - 3
\][/tex]
4. Write the Final Composed Function:
The composed function [tex]\( D \)[/tex] in terms of [tex]\( q \)[/tex] is:
[tex]\[
D = 72 q^9 - 3
\][/tex]
So, the composed function [tex]\( D \)[/tex] is:
[tex]\[
D = 72 q^9 - 3
\][/tex]
Therefore, by substitution, we have:
[tex]\[
D = 72 q^9 - 3
\][/tex]
