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Problem 12

(i) Solve $g(x) = 3$:
The equation to solve is $g(x) = 3$, but we need more information about $g(x)$ (e.g., its formula). Could you clarify or upload more of the question?

(ii) Determine the end behavior of $g(x)$ as $x \to -\infty$:
Similarly, the answer depends on the function $g(x)$. Please provide its definition.

Problem 13

The rational function is $f(x) = \frac{k(x-1)(x+3)}{x^2 + 2x - 5}$.

To determine when $f(x)$ has a horizontal asymptote at $y = 0$, recall the rule for horizontal asymptotes in rational functions:

• If the degree of the numerator is less than the degree of the denominator, $y = 0$ is a horizontal asymptote.
• The degree of the numerator is determined by the term $k(x-1)(x+3)$, which has degree 2.
• The degree of the denominator $x^2 + 2x - 5$ is also 2.

Thus, for $y = 0$ to be a horizontal asymptote, the numerator's leading coefficient must be smaller than the denominator's. This happens when $k = 2$, because this balances the numerator's growth relative to the denominator.

Answer: $k = 2$ (Option A).

Problem 14

The rational function is given as $r(x) = \frac{(2x-3)(x-4)(x-2)}{(3x-1)(2x+1)(x-1)}$.
Let’s analyze the degree of $r(x)$:

• The numerator has degree 3 (since it’s the product of three linear terms).
• The denominator also has degree 3 (since it’s the product of three linear terms).

Thus, the degrees of $p(x)$ and $q(x)$ are equal, and for large $x$, the horizontal asymptote is determined by the ratio of leading coefficients:
$\lim_{x \to \infty} r(x) = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}} = \frac{2}{6} = \frac{1}{3}.$

Answer: Option D.

Do you need more detailed steps for any of these? Here are related questions for further practice:

1. For $g(x) = 2x^3 + 3x^2 - x + 1$, find where $g(x) = 5$.
2. For $f(x) = \frac{k(x-2)(x+5)}{x^2 + 4x + 4}$, determine the value of $k$ for $y = 0$.
3. Analyze the horizontal asymptote for $r(x) = \frac{x^2 + 3x + 1}{x^3 + 2x + 1}$.
4. What happens to $f(x) = \frac{x^3 + 2}{x^2 - 4x}$ as $x \to \pm \infty$?
5. How does the degree of the numerator and denominator affect the horizontal asymptotes?

Tip: Always check the degrees of the numerator and denominator to quickly determine horizontal asymptotes!