find the equation of a rational function that passes through (0,1) and (2,1), has the x axis as a horizontal asymptote, and has two vertical asymptotes at x=3 and x=-3
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To find the equation of a rational function that satisfies the given conditions, we need to follow these steps:
Step 1: Determine the horizontal asymptote:
Since the x-axis is a horizontal asymptote, the denominator of the rational function should have a higher degree than the numerator.
Step 2: Identify the vertical asymptotes:
The vertical asymptotes are given as x = 3 and x = -3. Therefore, the rational function should have factors of (x - 3) and (x + 3) in the denominator.
Step 3: Determine the numerator:
To find the numerator, we need to consider the points (0,1) and (2,1), which the function passes through.
Let's assume the equation of the rational function is:
f(x) = (ax + b) / (x - 3)(x + 3)
Now, we can substitute the x-coordinate and y-coordinate of each point into the equation and solve for the unknowns a and b.
For (0, 1):
1 = (0a + b) / (0 - 3)(0 + 3)
1 = b / 9
b = 9
For (2, 1):
1 = (2a + 9) / (2 - 3)(2 + 3)
1 = (2a + 9) / (-1)(5)
1 = -(2a + 9) / 5
5 = -(2a + 9)
-2a - 9 = 5
-2a = 14
a = -7
Now, we substitute the values of a and b back into the equation:
f(x) = (-7x + 9) / (x - 3)(x + 3)
Therefore, the equation of the rational function that satisfies the given conditions is:
f(x) = (-7x + 9) / (x - 3)(x + 3)