if $c(x) = \frac{5}{x-2}$ and $d(x) = x + 3$, what is the domain of $(cd)(x)$?

all real values of $x$
all real values of $x$ except $x = 2$
all real values of $x$ except $x = -3$
all real values of $x$ except $x = 2$ and $x = -3$

Question

if $c(x) = \frac{5}{x-2}$ and $d(x) = x + 3$, what is the domain of $(cd)(x)$?

all real values of $x$
all real values of $x$ except $x = 2$
all real values of $x$ except $x = -3$
all real values of $x$ except $x = 2$ and $x = -3$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Function Multiplication" ], "new_concepts": [ "Domain of Combined Functions" ], "current_concepts": [ "Function Multiplication", "Domain of Combined Functions" ] } ## Identify individual domains Using the *Function Multiplication* knowledge point $$ \begin{aligned} &\text{For } c(x) = \frac{5}{x-2}: \quad x - 2 eq 0 \implies x eq 2\ &\text{For } d(x) = x + 3: \quad x \in \mathbb{R} \end{aligned} $$ ## Determine intersection of domains Using the *Function Multiplication* knowledge point $$ \begin{aligned} &\text{Domain of } (cd)(x) = \text{Domain}(c) \cap \text{Domain}(d)\ &x eq 2 \text{ and } x \in \mathbb{R} \implies x eq 2 \end{aligned} $$

all real values of $x$

all real values of $x$ except $x = 2$

all real values of $x$ except $x = -3$

all real values of $x$ except $x = 2$ and $x = -3$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Domain of Combined Functions" ] }

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QUESTION IMAGE

Question

Explanation:

Identify individual domains

Determine intersection of domains

Answer: