To find [tex]\((p \circ q)(x)\)[/tex], we need to determine the composition of the functions [tex]\(p(x)\)[/tex] and [tex]\(q(x)\)[/tex]. This means we will substitute [tex]\(q(x)\)[/tex] into [tex]\(p(x)\)[/tex].
### Step-by-Step Solution:
1. Given Functions:
- [tex]\(p(x) = 2x^2 - 4x\)[/tex]
- [tex]\(q(x) = x - 3\)[/tex]
2. Composition:
- We want to find [tex]\((p \circ q)(x)\)[/tex], which is [tex]\(p(q(x))\)[/tex]. This means we need to substitute [tex]\(q(x)\)[/tex] into [tex]\(p(x)\)[/tex].
3. Substitute [tex]\(q(x)\)[/tex] into [tex]\(p(x)\)[/tex]:
- First, find [tex]\(q(x)\)[/tex]:
[tex]\[
q(x) = x - 3
\][/tex]
- Substitute [tex]\(q(x)\)[/tex] into [tex]\(p\)[/tex]:
[tex]\[
p(q(x)) = p(x - 3)
\][/tex]
4. Evaluate [tex]\(p(x - 3)\)[/tex]:
- Start with the expression for [tex]\(p(x)\)[/tex]:
[tex]\[
p(x) = 2x^2 - 4x
\][/tex]
- Replace every [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]
5. Simplify [tex]\(p(x - 3)\)[/tex]:
- First, square the term [tex]\(x - 3\)[/tex]:
[tex]\[
(x - 3)^2 = x^2 - 6x + 9
\][/tex]
- Now multiply by 2:
[tex]\[
2(x^2 - 6x + 9) = 2x^2 - 12x + 18
\][/tex]
- Next, distribute [tex]\(-4\)[/tex] through [tex]\(x - 3\)[/tex]:
[tex]\[
-4(x - 3) = -4x + 12
\][/tex]
6. Combine the results:
- Add the two expressions we derived:
[tex]\[
2x^2 - 12x + 18 + (-4x + 12)
\][/tex]
- Simplify:
[tex]\[
2x^2 - 12x - 4x + 18 + 12 = 2x^2 - 16x + 30
\][/tex]
### Final Result:
Therefore, the composition [tex]\((p \circ q)(x) = 2x^2 - 16x + 30\)[/tex]. The correct answer is:
[tex]\[
\boxed{2x^2 - 16x + 30}
\][/tex]
