If [tex][tex]$p(x)=2x^2-4x$[/tex][/tex] and [tex][tex]$q(x)=x-3$[/tex][/tex], what is [tex][tex]$(p \circ q)(x)$[/tex][/tex]?

A. [tex]2x^2-4x+12[/tex]
B. [tex]2x^2-16x+18[/tex]
C. [tex]2x^2-16x+30[/tex]
D. [tex]2x^2-16x+15[/tex]



Answer :

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]