To write an equation for a rational function with the given properties, we need to consider the characteristics of the function and use that information to determine the equation.
a) For a hole at x = 1, we know that there will be a factor of (x - 1) in both the numerator and the denominator. The equation can be written as:
f(x) = (x - 1) / (x - 1) = 1, where (x - 1) in the numerator cancels out with (x - 1) in the denominator.
b) For a vertical asymptote anywhere and a horizontal asymptote along the x-axis, we know that the denominator must have a factor of x - a, where a is the x-coordinate of the vertical asymptote. The equation can be written as:
f(x) = 1 / (x - a), where a is the x-coordinate of the vertical asymptote.
c) For a hole at x = -2 and a vertical asymptote at x = 1, we need to include the factor of (x + 2) in both the numerator and the denominator, as well as the factor of (x - 1) in the denominator. The equation can be written as:
f(x) = (x + 2) / [(x + 2)(x - 1)], where (x + 2) in the numerator cancels out with (x + 2) in the denominator.
d) For a vertical asymptote at x = -1 and a horizontal asymptote at y = 2, we need the factor of (x + 1) in the denominator and a constant numerator equal to the horizontal asymptote value. The equation can be written as:
f(x) = 2 / (x + 1).
e) For an oblique asymptote without a vertical asymptote, we need to have a higher degree term in the numerator compared to the denominator. The equation can be written as:
f(x) = (4x^2 + 3x - 2) / (2x^2 + 5), where the numerator's degree is higher than the denominator's degree.
Note: These equations are just examples that satisfy the given properties. There may be multiple possible equations for each case, so the equations provided are not the only solutions.