Module 9: Rational Functions Assessment Review

A set of practice problems covering the simplification of rational expressions, identifying asymptotes, solving rational equations, and performing transformations on rational functions.

What is the simplified form of the expression $\frac{42x^2y^3}{105x^3}$?

$$\frac{2y^3}{5x}$$

How do you simplify the rational expression $\frac{2x^2-9x-5}{2x^2-11x+5}$?

Factor both the top and bottom to get $\frac{(x-5)(2x+1)}{(x-5)(2x-1)}$, which simplifies to $\frac{2x+1}{2x-1}$.

What is the least common multiple of $9x^2-4$ and $3x^2+13x-10$?

$$(3x-2)(3x+2)(x+5)$$

What is the result of simplifying $\frac{2}{3x^2} - \frac{5}{y}$ into a single rational expression?

$$\frac{2y-15x^2}{3x^2y}$$

Simplify the sum of $\frac{x^2-1}{x-5} + \frac{x+3}{x+2}$.

$$\frac{2x^2-2x-16}{(x-5)(x+2)}$$

What is the excluded value of $x$ for the function $f(x) = \frac{3}{3x-18}$?

$$x = 6$$

Describe the asymptotes and discontinuities for the function $f(x) = \frac{x^2-4}{x^2-x-2}$.

The graph has a horizontal asymptote at $y = 1$, a vertical asymptote at $x = -1$, and a point discontinuity at $x = 2$.

What are the solutions for $x$ in the equation $\frac{2}{x} + \frac{4}{x+3} = 3$?

$x = 7$ and $x = -2$

List all values of $x$ for which the original expression $\frac{9-x^2}{x^2-4x-21} \div \frac{2x^2+7x+3}{2x^2-15x+7}$ is restricted.

$$x = 7, -3, -\frac{1}{2}, \frac{1}{2}$$

Identify the domain and range for the function depicted as $f(x) = \frac{1}{x-4} - 3$.

Domain: $\{x | x \neq 4\}$; Range: $\{f(x) | f(x) \neq -3\}$

What are the values of $a, h, \text{ and } k$ for the function $f(x) = \frac{3}{x-4} - 2$?

$a = 3$, $h = 4$, and $k = -2$

Based on the function $f(x) = \frac{3}{x-4} - 2$, what transformations have occurred?

Vertical stretch by $3$, Shift to the Right $4$, and Shift Down $2$.

What is the solution for $x$ in the equation $\frac{5x-6}{x-2} = \frac{7}{3}$?

$$x = \frac{1}{2}$$

Solve the equation $\frac{6}{a+3} - \frac{8}{a-6} = \frac{14}{a^2-3a-18}$ for $a$.

$$a = 26$$