To solve for [tex]\( (p \circ q)(x) \)[/tex]:
First, we need to understand what [tex]\( (p \circ q)(x) \)[/tex] means. It represents the composition of the two functions [tex]\( p(x) \)[/tex] and [tex]\( q(x) \)[/tex], so we substitute [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]
The composition [tex]\( (p \circ q)(x) \)[/tex] is defined as [tex]\( p(q(x)) \)[/tex]. So, we substitute [tex]\( q(x) \)[/tex] into [tex]\( p(x) \)[/tex]:
1. Substitute [tex]\( q(x) = x - 3 \)[/tex] into [tex]\( p(x) \)[/tex]:
[tex]\[
p(q(x)) = p(x-3)
\][/tex]
2. Now, 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]
3. Simplify the expression:
[tex]\[
(x-3)^2 = x^2 - 6x + 9
\][/tex]
So,
[tex]\[
2(x-3)^2 = 2(x^2 - 6x + 9) = 2x^2 - 12x + 18
\][/tex]
And,
[tex]\[
-4(x-3) = -4x + 12
\][/tex]
4. Combine these results:
[tex]\[
p(x-3) = 2x^2 - 12x + 18 - 4x + 12
\][/tex]
Simplifying further,
[tex]\[
p(x-3) = 2x^2 - 16x + 30
\][/tex]
Thus, [tex]\( (p \circ q)(x) = 2x^2 - 16x + 30 \)[/tex].
Therefore, the correct answer is:
[tex]\[ 2 x^2 - 16 x + 30 \][/tex]
So, the correct option is:
[tex]\[ 2 x^2 - 16 x + 30 \][/tex]
