if $p(x) = 2x^2 - 4x$ and $q(x) = x - 3$, what is $(p \circ q)(x)$?

* $2x^2 - 4x + 12$
* $2x^2 - 16x + 18$
* $2x^2 - 16x + 30$
* $2x^2 - 16x + 15$

Question

if $p(x) = 2x^2 - 4x$ and $q(x) = x - 3$, what is $(p \circ q)(x)$?

* $2x^2 - 4x + 12$
* $2x^2 - 16x + 18$
* $2x^2 - 16x + 30$
* $2x^2 - 16x + 15$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Composite Functions" ], "new_concepts": [], "current_concepts": [ "Composite Functions", "Polynomial Operations" ] } ## Define the composite function $$ (p \circ q)(x) = p(q(x)) $$ ## Substitute q(x) into p(x) $$ \begin{aligned} p(q(x)) &= p(x - 3) \ &= 2(x - 3)^2 - 4(x - 3) \end{aligned} $$ ## Expand and simplify the expression $$ \begin{aligned} 2(x - 3)^2 - 4(x - 3) &= 2(x^2 - 6x + 9) - 4x + 12 \ &= 2x^2 - 12x + 18 - 4x + 12 \ &= 2x^2 - 16x + 30 \end{aligned} $$

(A) $2x^2 - 4x + 12$

(B) $2x^2 - 16x + 18$

(C) $2x^2 - 16x + 30$

(D) $2x^2 - 16x + 15$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Composite Functions" ] }

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QUESTION IMAGE

Question

Explanation:

Define the composite function

Substitute q(x) into p(x)

Expand and simplify the expression

Answer: