select the correct answer.

the domain of rational function $g$ is the same as the domain of rational function $f$. both $f$ and $g$ have a single $x$-intercept at $x = -10$. which equation could represent function $g$?

$g(x) = 10f(x)$
$g(x) = f(x + 10)$
$g(x) = f(x) + 10$
$g(x) = f(x) - 10$

Question

select the correct answer.

the domain of rational function $g$ is the same as the domain of rational function $f$. both $f$ and $g$ have a single $x$-intercept at $x = -10$. which equation could represent function $g$?

$g(x) = 10f(x)$
$g(x) = f(x + 10)$
$g(x) = f(x) + 10$
$g(x) = f(x) - 10$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Rational Functions" ], "new_concepts": [ "Function Transformations", "Domain and Intercepts" ], "current_concepts": [ "Rational Functions", "Function Transformations", "Domain and Intercepts" ] } ## Analyze the domain constraint $$ \text{Domain}(g) = \text{Domain}(f) \implies g(x) = c \cdot f(x) \text{ where } c eq 0 $$ ## Analyze the x-intercept constraint $$ \begin{aligned} &f(-10) = 0 \text{ and } g(-10) = 0\ &g(x) = 10f(x) \implies g(-10) = 10f(-10) = 10(0) = 0 \end{aligned} $$ ## Verify other options $$ \begin{aligned} &g(x) = f(x+10) \implies \text{Domain}(g) eq \text{Domain}(f)\ &g(x) = f(x) \pm 10 \implies g(-10) = f(-10) \pm 10 = \pm 10 eq 0 \end{aligned} $$

(A) $g(x) = 10f(x)$

(B) $g(x) = f(x + 10)$

(C) $g(x) = f(x) + 10$

(D) $g(x) = f(x) - 10$

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

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QUESTION IMAGE

Question

Explanation:

Analyze the domain constraint

Analyze the x-intercept constraint

Verify other options

Answer: