Of course! Let's go through the step-by-step process of composing the given functions using substitution.
We are given two functions:
[tex]\[ y = 5u^2 + 2u + 6 \][/tex]
[tex]\[ u = 2x^3 \][/tex]
We need to substitute [tex]\( u \)[/tex] into the function [tex]\( y \)[/tex].
1. Start by substituting [tex]\( u \)[/tex] from the second function into the first function:
[tex]\[ y = 5(2x^3)^2 + 2(2x^3) + 6 \][/tex]
2. Simplify the expression inside the parentheses:
[tex]\[ (2x^3)^2 = 4x^6 \][/tex]
So the equation becomes:
[tex]\[ y = 5(4x^6) + 2(2x^3) + 6 \][/tex]
3. Next, simplify each term:
[tex]\[ 5(4x^6) = 20x^6 \][/tex]
[tex]\[ 2(2x^3) = 4x^3 \][/tex]
4. Combine all the simplified terms:
[tex]\[ y = 20x^6 + 4x^3 + 6 \][/tex]
Therefore, the composed function [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex] is:
[tex]\[
y = 20x^6 + 4x^3 + 6
\][/tex]
