Find a formula for a function that has vertical asymptotes at x=7 and x=9 and a horizontal asymptote at y=7

how about

y = 7x^2/((x-7)(x-9))

Well, let's see... If we want vertical asymptotes at x=7 and x=9, we can use the factors (x-7) and (x-9) in the denominator. As for the horizontal asymptote at y=7, we'll need a constant value in the numerator.

So, using these ingredients, we can come up with a formula like this:

f(x) = 7 / ((x-7)(x-9))

This function would have vertical asymptotes at x=7 and x=9, and a horizontal asymptote at y=7. Keep in mind, though, that there are many other possible formulas that could also meet these requirements!

To find a formula for a function with vertical asymptotes at x=7 and x=9 and a horizontal asymptote at y=7, we can start by considering the general form of a rational function.

A rational function can be defined as:

f(x) = (P(x))/(Q(x))

where P(x) and Q(x) are polynomials.

For vertical asymptotes at x=7 and x=9, we need to ensure that the denominator Q(x) has zeros at x=7 and x=9. This will cause the function to approach infinity as x gets close to these values.

Therefore, let's assume that the denominator Q(x) can be factored as follows:

Q(x) = (x - 7)(x - 9)

Now, for a horizontal asymptote at y=7, we want the degree of the numerator P(x) to be less than or equal to the degree of the denominator Q(x). This will ensure that the function approaches the horizontal line y=7 as x goes to infinity or negative infinity.

To satisfy this condition, let's assume that the numerator P(x) is a constant, let's say k.

Therefore, our function becomes:

f(x) = k/((x - 7)(x - 9))

In this form, we have a function with vertical asymptotes at x=7 and x=9 and a horizontal asymptote at y=7. The specific value of k will determine the behavior of the function at other points.

To find a formula for a function with the given vertical and horizontal asymptotes, we can start by constructing the general form of the rational function in the following way:

Let's assume our function is f(x). Since we have vertical asymptotes at x = 7 and x = 9, there will be factors of (x - 7) and (x - 9) in the denominator to cancel out these vertical asymptotes.

Now, since we have a horizontal asymptote at y = 7, the numerator of our function should not have any terms that increase in degree faster than the denominator.

Putting all these conditions together, we can write the formula for our function as:

f(x) = c(x - 7)(x - 9) / (x - a)(x - b) + 7

where c is a constant and a, b are any other distinct values.

This formula satisfies the requirements of having vertical asymptotes at x = 7 and x = 9, a horizontal asymptote at y = 7, and being a rational function.

Keep in mind that depending on specific constraints or additional information, the values of c, a, and b may need to be adjusted to fit the given conditions.