To find the composition of the functions [tex]\(p(x)\)[/tex] and [tex]\(q(x)\)[/tex], denoted as [tex]\((p \circ q)(x)\)[/tex], follow these steps:
1. Define the given functions:
- [tex]\( p(x) = 2x^2 - 4x \)[/tex]
- [tex]\( q(x) = x - 3 \)[/tex]
2. Substitute [tex]\(q(x)\)[/tex] into [tex]\(p(x)\)[/tex]:
This means we need to evaluate [tex]\(p\)[/tex] at [tex]\(q(x)\)[/tex], or [tex]\( p(q(x)) \)[/tex]. Substitute [tex]\(q(x) = x - 3\)[/tex] into [tex]\(p(x)\)[/tex]:
[tex]\[
p(q(x)) = p(x - 3)
\][/tex]
3. Replace [tex]\(x\)[/tex] in [tex]\(p(x)\)[/tex] with [tex]\(x - 3\)[/tex]:
[tex]\[
p(x - 3) = 2(x - 3)^2 - 4(x - 3)
\][/tex]
4. Expand and simplify:
- First, expand [tex]\((x - 3)^2\)[/tex]:
[tex]\[
(x - 3)^2 = x^2 - 6x + 9
\][/tex]
- Substitute back in:
[tex]\[
p(x - 3) = 2(x^2 - 6x + 9) - 4(x - 3)
\][/tex]
5. Distribute and combine like terms:
- Distribute [tex]\(2\)[/tex] in the first term:
[tex]\[
= 2x^2 - 12x + 18
\][/tex]
- Distribute [tex]\(-4\)[/tex] in the second term:
[tex]\[
= -4x + 12
\][/tex]
- Combine both results:
[tex]\[
p(x - 3) = 2x^2 - 12x + 18 - 4x + 12
\][/tex]
- Combine like terms:
[tex]\[
2x^2 - 16x + 30
\][/tex]
Hence, [tex]\((p \circ q)(x) = 2x^2 - 16x + 30\)[/tex].
Among the given choices, the correct answer is:
[tex]\[
\boxed{2x^2 - 16x + 30}
\][/tex]
