if $f(x) = x + 7$ and $g(x) = \frac{1}{x - 13}$, what is the domain of $(f \circ g)(x)$?

$\circ$ $\{x \mid x \
eq 6\}$
$\circ$ $\{x \mid x \
eq -6\}$
$\circ$ $\{x \mid x \
eq -13\}$
$\circ$ $\{x \mid x \
eq 13\}$

Question

if $f(x) = x + 7$ and $g(x) = \frac{1}{x - 13}$, what is the domain of $(f \circ g)(x)$?

$\circ$ $\{x \mid x \
eq 6\}$
$\circ$ $\{x \mid x \
eq -6\}$
$\circ$ $\{x \mid x \
eq -13\}$
$\circ$ $\{x \mid x \
eq 13\}$

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", "Domain of Composite Functions" ], "new_concepts": [], "current_concepts": [ "Composite Functions", "Domain of Composite Functions" ] } ## Identify the inner function and its domain $$ \begin{aligned} &g(x) = \frac{1}{x - 13} \ &x - 13 eq 0 \implies x eq 13 \end{aligned} $$ ## Identify the outer function and its domain $$ \begin{aligned} &f(x) = x + 7 \ &\text{Domain of } f(x) \text{ is all real numbers } \mathbb{R} \end{aligned} $$ ## Determine the domain of the composite function $$ \begin{aligned} &\text{Domain of } (f \circ g)(x) = \{x \in \text{Domain of } g \mid g(x) \in \text{Domain of } f\} \ &x eq 13 \text{ and } \frac{1}{x - 13} \in \mathbb{R} \implies x eq 13 \end{aligned} $$

(A) $\{x \mid x eq 6\}$

(B) $\{x \mid x eq -6\}$

(C) $\{x \mid x eq -13\}$

(D) $\{x \mid x eq 13\}$

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

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QUESTION IMAGE

Question

Explanation:

Identify the inner function and its domain

Identify the outer function and its domain

Determine the domain of the composite function

Answer: