To determine [tex]\((p \circ q)(x)\)[/tex], we need to find [tex]\(p(q(x))\)[/tex], which involves substituting [tex]\(q(x)\)[/tex] into [tex]\(p(x)\)[/tex].
Given the functions:
[tex]\[ p(x) = 2x^2 - 4x \][/tex]
[tex]\[ q(x) = x - 3 \][/tex]
First, we substitute [tex]\( q(x) \)[/tex] into [tex]\( p(x) \)[/tex]:
[tex]\[ p(q(x)) = p(x-3) \][/tex]
This means we 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]
Now, let's break this down step-by-step:
1. Calculate [tex]\((x-3)^2\)[/tex]:
[tex]\[ (x-3)^2 = x^2 - 6x + 9 \][/tex]
2. Multiply this result by 2:
[tex]\[ 2(x-3)^2 = 2(x^2 - 6x + 9) = 2x^2 - 12x + 18 \][/tex]
3. Calculate [tex]\(-4(x-3)\)[/tex]:
[tex]\[ -4(x-3) = -4x + 12 \][/tex]
4. Combine these results:
[tex]\[ p(x-3) = 2x^2 - 12x + 18 - 4x + 12 \][/tex]
[tex]\[ p(x-3) = 2x^2 - 12x - 4x + 18 + 12 \][/tex]
[tex]\[ p(x-3) = 2x^2 - 16x + 30 \][/tex]
Therefore, [tex]\((p \circ q)(x) = 2x^2 - 16x + 30\)[/tex].
So, looking at the options provided:
[tex]\[ 2 x^2 - 4 x + 12 \][/tex]
[tex]\[ 2 x^2 - 16 x + 18 \][/tex]
[tex]\[ 2 x^2 - 16 x + 30 \][/tex]
[tex]\[ 2 x^2 - 16 x + 15 \][/tex]
The correct answer is:
[tex]\[ 2 x^2 - 16 x + 30 \][/tex]
