select the correct answer.

the domain of function $r$ is $(-\infty, -7) \cup (-7, 5) \cup (5, \infty)$, and the value of the function approaches 1 as $x$ approaches $-\infty$ and $\infty$. which function could be function $r$?

$f(x) = \frac{x - 5}{x + 7}$

$f(x) = \frac{2x - 35}{x^2 + 2x - 35}$

$f(x) = \frac{x^2 - 3x - 15}{x^2 + 2x - 35}$

$f(x) = \frac{x^2 + 2x - 35}{x^2 - 3x - 15}$

Question

select the correct answer.

the domain of function $r$ is $(-\infty, -7) \cup (-7, 5) \cup (5, \infty)$, and the value of the function approaches 1 as $x$ approaches $-\infty$ and $\infty$. which function could be function $r$?

$f(x) = \frac{x - 5}{x + 7}$

$f(x) = \frac{2x - 35}{x^2 + 2x - 35}$

$f(x) = \frac{x^2 - 3x - 15}{x^2 + 2x - 35}$

$f(x) = \frac{x^2 + 2x - 35}{x^2 - 3x - 15}$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Domain of a Function", "Rational Functions", "Asymptotes" ], "new_concepts": [], "current_concepts": [ "Domain of a Function", "Rational Functions", "Asymptotes" ] } ## Analyze the domain constraints $$ \begin{aligned} &\text{Domain: } (-\infty, -7) \cup (-7, 5) \cup (5, \infty)\ &\text{Excluded values: } x = -7 \text{ and } x = 5\ &\text{Denominator must be zero at: } x = -7 \text{ and } x = 5\ &D(x) = (x + 7)(x - 5) = x^2 + 2x - 35 \end{aligned} $$ ## Analyze the horizontal asymptote $$ \begin{aligned} &\lim_{x \to \pm\infty} f(x) = 1\ &\text{Since the limit is 1, the degrees of numerator and denominator must be equal,}\ &\text{and the ratio of their leading coefficients must be 1.}\ &\text{Denominator degree is 2, so numerator degree must be 2 with leading coefficient 1.} \end{aligned} $$ ## Identify the matching function $$ \begin{aligned} &\text{Denominator: } x^2 + 2x - 35\ &\text{Numerator: } x^2 - 3x - 10 \quad (\text{degree 2, leading coefficient 1})\ &f(x) = \frac{x^2 - 3x - 10}{x^2 + 2x - 35} \end{aligned} $$

(A) $f(x) = \frac{x - 5}{x + 7}$

(B) $f(x) = \frac{2x - 35}{x^2 + 2x - 35}$

(C) $f(x) = \frac{x^2 - 3x - 10}{x^2 + 2x - 35}$

(D) $f(x) = \frac{x^2 + 2x - 35}{x^2 - 3x - 10}$

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

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QUESTION IMAGE

Question

Explanation:

Analyze the domain constraints

Analyze the horizontal asymptote

Identify the matching function

Answer: