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Find a function that has vertical asymptotes at x=-4 and x=4 and a horizontal asymptote at y=1.

1 Answer

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To find such a function, we follow these steps:

1. Recognizing the form of the function:
The function can be a rational function in the format of (ax + b) / ((x - c)(x - d)).

2. Identifying the vertical asymptotes:
For vertical asymptotes at x = -4 and x = 4, the denominator of our function must become zero at these points, hence, c = -4 and d = 4. From this, we can express the denominator as (x - (-4))(x - 4) = (x + 4)(x - 4) = x^2 - 16.

3. Identifying the horizontal asymptote:
For a horizontal asymptote at y = 1, the degree of the numerator and denominator must be equal, and the coefficients of the highest-degree terms must also be equal. The highest degree term in our denominator is x^2, so the numerator must also be a quadratic in the form ax^2 + bx + c. Since the coefficients of the highest-degree terms must be equal, a = 1.

4. Determining the format of the function:
Therefore, the function is of the form (x^2 + bx + c) / (x^2 - 16) = 1.

5. Solving for b and c:
Solving for b and c, we find that b = 0 and c = 16.

So, the function that satisfies the given condition is f(x) = (x^2 + 16) / (x^2 - 16).

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