given: $f(x) = \frac{1}{x+5}$ and $g(x) = x - 2$.
what are the restrictions of the domain of $f(g(x))$?

$x \
eq -5$
$x \
eq -3$
$x \
eq 2$
there are no restrictions.

Question

given: $f(x) = \frac{1}{x+5}$ and $g(x) = x - 2$.
what are the restrictions of the domain of $f(g(x))$?

$x \
eq -5$
$x \
eq -3$
$x \
eq 2$
there are no restrictions.

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": [ "Composite Functions" ], "new_concepts": [ "Domain of Composite Functions" ], "current_concepts": [ "Composite Functions", "Domain of Composite Functions" ] } ## Identify domain restrictions of the inner function Using the *Composite Functions* knowledge point $$ \begin{aligned} &g(x) = x - 2 \ &\text{Domain of } g(x): x \in \mathbb{R} \quad (\text{no restrictions}) \end{aligned} $$ ## Determine restrictions from the outer function Using the *Composite Functions* knowledge point $$ \begin{aligned} &f(u) = \frac{1}{u+5} \implies u e -5 \ &g(x) e -5 \implies x - 2 e -5 \ &x e -3 \end{aligned} $$

(A) $x eq -5$

(B) $x eq -3$

(C) $x eq 2$

(D) There are no restrictions.

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

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Identify domain restrictions of the inner function

Determine restrictions from the outer function

Answer: