What is the general form of a rational function?
$r(x) = \frac{P(x)}{Q(x)}$, where P(x) and Q(x) are polynomials.
How to find real zeros of $r(x) = \frac{P(x)}{Q(x)}$?
Solve $P(x) = 0$, ensuring that the solutions are not zeros of $Q(x)$.
How do you represent interval analysis?
$(-\infty, a)$, $(a, b)$, $(b, \infty)$ where a and b are critical points.
What is the condition for a vertical asymptote at $x=a$?
$Q(a) = 0$ and $P(a) \ne 0$ for $r(x) = \frac{P(x)}{Q(x)}$.
Formula for solving inequalities with rational functions?
Analyze the sign of $r(x)$ in intervals defined by zeros and vertical asymptotes.
What is the factored form of a quadratic?
$ax^2 + bx + c = a(x - r_1)(x - r_2)$, where $r_1$ and $r_2$ are the roots.
How do you find the domain of $r(x) = \frac{P(x)}{Q(x)}$?
Domain = {$x \in \mathbb{R} | Q(x) \ne 0$}
How do you simplify a rational function?
$\frac{P(x)}{Q(x)} = \frac{(x-a)R(x)}{(x-a)S(x)} = \frac{R(x)}{S(x)}$, if $x \ne a$.
What is the form of a linear equation?
$y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
How do you represent a rational function with a removable discontinuity?
$r(x) = \frac{(x-a)P(x)}{(x-a)}$, where the discontinuity is at $x=a$.
What does a zero on the graph of a rational function represent?
It represents an x-intercept, where the function's value is zero.
What does a vertical asymptote on the graph of a rational function represent?
It represents a point where the function is undefined and approaches infinity or negative infinity.
How can you identify intervals where a rational function is positive from its graph?
These are the intervals where the graph is above the x-axis.
How can you identify intervals where a rational function is negative from its graph?
These are the intervals where the graph is below the x-axis.
What does a hole in the graph of a rational function represent?
It represents a removable discontinuity, where a factor in the numerator and denominator cancels out.
How does the graph of $r(x)$ behave near a vertical asymptote?
The graph approaches infinity or negative infinity as $x$ approaches the asymptote.
What does it mean if the graph of $r(x)$ crosses the x-axis at $x=a$?
It means $x=a$ is a real zero of the rational function.
How do you identify the domain from the graph of a rational function?
The domain consists of all x-values except those at vertical asymptotes or holes.
What does the end behavior of the graph of a rational function tell you?
It describes how the function behaves as x approaches positive or negative infinity.
How can you tell if a rational function has a horizontal asymptote from its graph?
If the graph approaches a constant y-value as x goes to positive or negative infinity.
How to find real zeros of $f(x) = \frac{x^2 - 1}{x + 1}$?
1. Factor: $f(x) = \frac{(x-1)(x+1)}{x+1}$. 2. Domain: $x \ne -1$. 3. Simplify: $f(x) = x-1, x \ne -1$. 4. Numerator zero: $x = 1$.
How to solve $\frac{x - 2}{x + 3} > 0$?
1. Critical points: $x = 2, x = -3$. 2. Intervals: $(-\infty, -3), (-3, 2), (2, \infty)$. 3. Test values. 4. Solution: $(-\infty, -3) \cup (2, \infty)$.
How to find vertical asymptote(s) of $f(x) = \frac{x}{x - 5}$?
1. Denominator zero: $x - 5 = 0$. 2. Solve: $x = 5$. 3. Check numerator: $f(5)$ is undefined. 4. Vertical asymptote: $x = 5$.
How to determine where $f(x) = \frac{x + 1}{x - 2}$ is negative?
1. Critical points: $x = -1, x = 2$. 2. Intervals: $(-\infty, -1), (-1, 2), (2, \infty)$. 3. Test values. 4. Solution: $(-1, 2)$.
Find the domain of $f(x) = \frac{x^2 - 4}{x - 2}$?
1. Denominator: $x - 2$. 2. Set $x - 2 \ne 0$. 3. Solve: $x \ne 2$. 4. Domain: All real numbers except 2.
How to find the real zeros of $r(x) = \frac{x^2 - 5x + 6}{x - 1}$?
1. Factor the numerator: $x^2 - 5x + 6 = (x-2)(x-3)$. 2. Set numerator to zero: $(x-2)(x-3) = 0$. 3. Solve for x: $x = 2, 3$. 4. Check domain: $x \ne 1$. 5. Zeros: $x = 2, 3$.
How to solve the inequality $\frac{x + 4}{x - 1} \le 0$?
1. Find critical points: $x = -4, x = 1$. 2. Create intervals: $(-\infty, -4], [-4, 1), (1, \infty)$. 3. Test values in each interval. 4. Include endpoints where function equals zero. 5. Solution: $[-4, 1)$.
How do you find the vertical asymptote of $f(x) = \frac{x + 2}{x^2 - 9}$?
1. Factor the denominator: $x^2 - 9 = (x - 3)(x + 3)$. 2. Set denominator equal to zero: $(x - 3)(x + 3) = 0$. 3. Solve for x: $x = 3, -3$. 4. Check numerator at these points. 5. Vertical asymptotes: $x = 3, x = -3$.
How do you determine the intervals where $f(x) = \frac{x^2 - 16}{x + 2}$ is positive?
1. Factor the numerator: $x^2 - 16 = (x - 4)(x + 4)$. 2. Find critical points: $x = -4, 4, -2$. 3. Create intervals: $(-\infty, -4), (-4, -2), (-2, 4), (4, \infty)$. 4. Test values in each interval. 5. Solution: $(-4, -2) \cup (4, \infty)$.
How do you find the domain of $f(x) = \frac{x^2 - 1}{x^2 - 4x + 3}$?
1. Factor the denominator: $x^2 - 4x + 3 = (x - 1)(x - 3)$. 2. Set denominator not equal to zero: $(x - 1)(x - 3) \ne 0$. 3. Solve for x: $x \ne 1, 3$. 4. Domain: All real numbers except 1 and 3.
