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: $x = 1$.

1. Critical points: $x = 2, x = -3$. 2. Intervals: $(-\infty, -3), (-3, 2), (2, \infty)$. 3. Test values. 4. Solution: $(-\infty, -3) \cup (2, \infty)$.

1. Denominator zero: $x - 5 = 0$. 2. Solve: $x = 5$. 3. Check numerator: $f(5)$ is undefined. 4. Vertical asymptote: $x = 5$.

1. Critical points: $x = -1, x = 2$. 2. Intervals: $(-\infty, -1), (-1, 2), (2, \infty)$. 3. Test values. 4. Solution: $(-1, 2)$.

1. Denominator: $x - 2$. 2. Set $x - 2 \ne 0$. 3. Solve: $x \ne 2$. 4. Domain: All real numbers except 2.

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: $x = 2, 3$.

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: $[-4, 1)$.

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: $x = 3, x = -3$.

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: $(-4, -2) \cup (4, \infty)$.

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.

$r(x) = \frac{P(x)}{Q(x)}$, where P(x) and Q(x) are polynomials.

Solve $P(x) = 0$, ensuring that the solutions are not zeros of $Q(x)$.

$(-\infty, a)$, $(a, b)$, $(b, \infty)$ where a and b are critical points.

$Q(a) = 0$ and $P(a) \ne 0$ for $r(x) = \frac{P(x)}{Q(x)}$.

Analyze the sign of $r(x)$ in intervals defined by zeros and vertical asymptotes.

$ax^2 + bx + c = a(x - r_1)(x - r_2)$, where $r_1$ and $r_2$ are the roots.

Domain = {$x \in \mathbb{R} | Q(x) \ne 0$}

$\frac{P(x)}{Q(x)} = \frac{(x-a)R(x)}{(x-a)S(x)} = \frac{R(x)}{S(x)}$, if $x \ne a$.

$y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

$r(x) = \frac{(x-a)P(x)}{(x-a)}$, where the discontinuity is at $x=a$.

Zeros: function equals zero | Vertical Asymptotes: function is undefined

Removable: factor cancels out, hole in graph | Non-removable: vertical asymptote

$r(x) = 0$: find zeros | $r(x) > 0$: interval analysis

Numerator: determines zeros | Denominator: determines asymptotes

Polynomial: all real numbers | Rational: excludes zeros of denominator

Endpoints: function is zero | Asymptotes: function is undefined

$r(x) > 0$: find intervals where function is positive | $r(x) < 0$: find intervals where function is negative

Near a zero: the function crosses or touches the x-axis | Near a vertical asymptote: the function approaches infinity or negative infinity

Graphically: identify x-intercepts | Algebraically: solve for x when the function equals zero

Rational equations: find specific values | Rational inequalities: find intervals