1. f(x) = 1/(x - 3):
Domain: All real numbers except x = 3.
Intercepts: None.
Asymptotes: Vertical asymptote at x = 3.
Additional Points: Choose x-values on both sides of 3, e.g., x = 2, 4.
2. f(x) = x²/(x² + 9):
Domain: All real numbers.
Intercepts: Y-intercept at (0, 0).
Asymptotes: None.
Additional Points: Choose x-values and find corresponding y-values.
3. f(t) = (1 - 2t)/(t² - 3t - 4):
Domain: All real numbers except t = -1, 4.
Intercepts: Y-intercept at (0, 1).
Asymptotes: None.
Additional Points: Choose x-values and find corresponding y-values.
4. g(s) = 4s/(s² + 4):
Domain: All real numbers.
Intercepts: Y-intercept at (0, 0).
Asymptotes: None.
Additional Points: Choose x-values and find corresponding y-values.
5. f(x) = -x/(x - 2)²:
Domain: All real numbers except x = 2.
Intercepts: X-intercept at (0, 0).
Asymptotes: Vertical asymptote at x = 2.
Additional Points: Choose x-values on both sides of 2.
6. h(x) = 2x/(x² - 3x - 4):
Domain: All real numbers except x = -1, 4.
Intercepts: Y-intercept at (0, 0).
Asymptotes: None.
Additional Points: Choose x-values and find corresponding y-values.
1. f(x) = 1/(x - 3):
Domain: The domain is all real numbers except the value that makes the denominator zero. In this case, the denominator is (x - 3), so x ≠ 3.
Intercepts: There are no intercepts in this case as the numerator is always 1.
Asymptotes: There is a vertical asymptote at x = 3 since division by zero occurs at this point.
2. f(x) = x²/(x² + 9):
Domain: There are no restrictions on the domain since there are no denominators.
Intercepts: The y-intercept is found by setting x = 0, giving (0, 0).
Asymptotes: Since there are no denominators, there are no vertical or horizontal asymptotes.
3. f(t) = (1 - 2t)/(t² - 3t - 4):
Domain: Exclude values making the denominator zero. Factor the denominator: (t - 4)(t + 1). Set each factor equal to zero to find t ≠ -1, 4.
Intercepts: The y-intercept is found by setting t = 0, giving (0, 1).
Asymptotes: There are no vertical or horizontal asymptotes.
4. g(s) = 4s/(s² + 4):
Domain: No restrictions on the domain.
Intercepts: The y-intercept is found by setting s = 0, giving (0, 0).
Asymptotes: No asymptotes since the degree of the numerator is the same as the degree of the denominator.
5. f(x) = -x/(x - 2)²:
Domain: Exclude x = 2, as it makes the denominator zero.
Intercepts: The x-intercept is found by setting y = 0 and solving for x.
Asymptotes: Vertical asymptote at x = 2 due to division by zero.
6. h(x) = 2x/(x² - 3x - 4):
Domain: Exclude x = -1, 4, as they make the denominator zero.
Intercepts: The y-intercept is found by setting x = 0, giving (0, 0).
Asymptotes: No asymptotes.