a company has determined that its weekly profit is a function of the number of items that it sells. which equation could represent the weekly profit in thousands of dollars, $y$, when the company sells $x$ items?

$y^2 = 4x^2 - 100$
$y = -x^2 + 50x - 300$
$x = -y^2 + 60y - 400$
$x^2 = -6y^2 + 200$

Question

a company has determined that its weekly profit is a function of the number of items that it sells. which equation could represent the weekly profit in thousands of dollars, \$y\$, when the company sells \$x\$ items?

\$y^2 = 4x^2 - 100\$
\$y = -x^2 + 50x - 300\$
\$x = -y^2 + 60y - 400\$
\$x^2 = -6y^2 + 200\$

$y^2 = 4x^2 - 100$
$y = -x^2 + 50x - 300$
$x = -y^2 + 60y - 400$
$x^2 = -6y^2 + 200$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Function Definition", "Quadratic Functions" ], "current_concepts": [ "Function Definition", "Quadratic Functions" ] } ## Identify the dependent and independent variables $$ \begin{aligned} &\text{Independent variable (input): } x \text{ (number of items sold)} \ &\text{Dependent variable (output): } y \text{ (weekly profit)} \end{aligned} $$ ## Apply the definition of a function $$ \begin{aligned} &y \text{ is a function of } x \implies y = f(x) \ &\text{For each input } x, \text{ there must be exactly one output } y. \end{aligned} $$ ## Evaluate the given options $$ \begin{aligned} &y^2 = 4x^2 - 100 \implies y = \pm\sqrt{4x^2 - 100} \quad (\text{not a function of } x) \ &y = -x^2 + 50x - 300 \quad (\text{expresses } y \text{ uniquely in terms of } x) \ &x = -y^2 + 60y - 400 \implies y \text{ is not a function of } x \ &x^2 = -6y^2 + 200 \implies y = \pm\sqrt{\frac{200 - x^2}{6}} \quad (\text{not a function of } x) \end{aligned} $$

(A) $y^2 = 4x^2 - 100$

(B) $y = -x^2 + 50x - 300$

(C) $x = -y^2 + 60y - 400$

(D) $x^2 = -6y^2 + 200$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Function Definition" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the dependent and independent variables

Apply the definition of a function

Evaluate the given options

Answer: