To find [tex]\((p \circ q)(x)\)[/tex], we need to substitute [tex]\(q(x)\)[/tex] into [tex]\(p(x)\)[/tex]. In other words, we need to find [tex]\(p(q(x))\)[/tex].
Given:
[tex]\[ p(x) = 2x^2 - 4x \][/tex]
[tex]\[ q(x) = x - 3 \][/tex]
We need to substitute [tex]\(x - 3\)[/tex] wherever we see [tex]\(x\)[/tex] in [tex]\(p(x)\)[/tex].
First, let's rewrite [tex]\(p(x)\)[/tex] with [tex]\(q(x)\)[/tex] inside it:
[tex]\[ p(q(x)) = p(x - 3) \][/tex]
Now, substitute [tex]\(x - 3\)[/tex] into [tex]\(p(x)\)[/tex]:
[tex]\[ p(x - 3) = 2(x - 3)^2 - 4(x - 3) \][/tex]
Next, we need to expand and simplify [tex]\(2(x - 3)^2 - 4(x - 3)\)[/tex].
First, expand [tex]\((x - 3)^2\)[/tex]:
[tex]\[ (x - 3)^2 = x^2 - 6x + 9 \][/tex]
Then multiply by 2:
[tex]\[ 2(x - 3)^2 = 2(x^2 - 6x + 9) = 2x^2 - 12x + 18 \][/tex]
Next, distribute the [tex]\(-4\)[/tex]:
[tex]\[ -4(x - 3) = -4x + 12 \][/tex]
Combine the two parts:
[tex]\[ 2x^2 - 12x + 18 - 4x + 12 \][/tex]
Combine like terms:
[tex]\[ 2x^2 - 12x - 4x + 18 + 12 = 2x^2 - 16x + 30 \][/tex]
Therefore, [tex]\((p \circ q)(x) = 2x^2 - 16x + 30\)[/tex].
Thus, the correct answer is:
[tex]\[ 2 x^2 - 16 x + 30 \][/tex]
