if $y \propto \frac{1}{x}$ and $y = -2$ when $x = 14$, find the equation that connects $x$ and $y$.

a) $y = -\frac{7}{x}$
b) $y = -7x$
c) $y = -28x$
d) $y = -\frac{28}{x}$

question 18 (5 points)
multiply the rational expressions: $\frac{x^2 - x}{(x + 1)^2} \cdot \frac{x^2 - 1}{x}$

a) $\frac{(x - 1)^2}{x(x + 1)}$
b) $\frac{x - 1}{x + 1}$
c) $\frac{(x - 1)^2}{x + 1}$
d) $\frac{1}{x + 1}$

question 19 (5 points)
determine any horizontal or slant asymptotes of the rational function: $f(x) = \frac{3x^2 - x}{2x^2 - 1}$

a) $f(x)$ has a slant asymptote at $y = x + \frac{3}{2}$.
b) $f(x)$ has a horizontal asymptote at $y = \frac{3}{2}$.
c) $f(x)$ has a horizontal asymptote at $y = 0$.
d) $f(x)$ has a slant asymptote at $y = \frac{3}{2}$.

Question

if $y \propto \frac{1}{x}$ and $y = -2$ when $x = 14$, find the equation that connects $x$ and $y$.

a) $y = -\frac{7}{x}$
b) $y = -7x$
c) $y = -28x$
d) $y = -\frac{28}{x}$

question 18 (5 points)
multiply the rational expressions: $\frac{x^2 - x}{(x + 1)^2} \cdot \frac{x^2 - 1}{x}$

a) $\frac{(x - 1)^2}{x(x + 1)}$
b) $\frac{x - 1}{x + 1}$
c) $\frac{(x - 1)^2}{x + 1}$
d) $\frac{1}{x + 1}$

question 19 (5 points)
determine any horizontal or slant asymptotes of the rational function: $f(x) = \frac{3x^2 - x}{2x^2 - 1}$

a) $f(x)$ has a slant asymptote at $y = x + \frac{3}{2}$.
b) $f(x)$ has a horizontal asymptote at $y = \frac{3}{2}$.
c) $f(x)$ has a horizontal asymptote at $y = 0$.
d) $f(x)$ has a slant asymptote at $y = \frac{3}{2}$.

Accepted Answer

{ "quality": "clear", "question_count": 3, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Inverse Variation", "Rational Expression Simplification", "Slant Asymptotes" ], "new_concepts": [], "current_concepts": [ "Inverse Variation", "Rational Expression Simplification", "Slant Asymptotes" ] } ## Solve the inverse variation equation $$ \begin{aligned} &y = \frac{k}{x} \implies -2 = \frac{k}{14} \implies k = -28\ &y = -\frac{28}{x} \end{aligned} $$ ## Simplify the rational product $$ \begin{aligned} &\frac{x^2-x}{(x+1)^2} \cdot \frac{x^2-1}{x} = \frac{x(x-1)}{(x+1)^2} \cdot \frac{(x-1)(x+1)}{x}\ &=\frac{x(x-1)(x-1)(x+1)}{x(x+1)^2} = \frac{(x-1)^2}{x+1} \end{aligned} $$ ## Determine the asymptotes of the rational function $$ \begin{aligned} &f(x) = \frac{3x^2-x}{2x^2-1}\ &\lim_{x \to \pm\infty} f(x) = \lim_{x \to \pm\infty} \frac{3x^2}{2x^2} = \frac{3}{2}\ &\text{Horizontal asymptote at } y = \frac{3}{2} \end{aligned} $$ ## Question 1 (A) $y = -7/x$

(B) $y = -7x$

(C) $y = -28x$

(D) $y = -28/x$ ## Question 18 (A) $\frac{(x-1)^2}{x(x+1)}$

(B) $\frac{x-1}{x+1}$

(C) $\frac{(x-1)^2}{x+1}$

(D) $\frac{1}{x+1}$ ## Question 19 (A) $f(x)$ has a slant asymptote at $y = x + 3/2$.

(B) $f(x)$ has a horizontal asymptote at $y = 3/2$.

(C) $f(x)$ has a horizontal asymptote at $y = 0$.

(D) $f(x)$ has a slant asymptote at $y = 3/2$.

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

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QUESTION IMAGE

Question

Explanation:

Solve the inverse variation equation

Simplify the rational product

Determine the asymptotes of the rational function

Answer:

Question 1

Question 18

Question 19