There are an infinite number of correct answers to this.
Let's assume that we are creating a RATIONAL function (polynomials on top and bottom).
First, let's ensure that our fraction yields 0 / something when x = 3. Also, since we have an even function, we need to get 0 when x = -3 as well. We can do this by putting an (x - 3)(x + 3) as a factor in the numerator.
Secondly, we need to ensure that we have c / 0 (c≠0) when x = 4 and x = - 4. We can accomplish this by choosing to put (x - 4) and (x + 4) in the denominator.
So far we have:
(x² - 9) / ((x - 4)(x + 4))
or
(x² - 9) / (x² - 16)
To get the rest of the details right, let's multiply the function by (Ax² + B)/(Cx² + D). We could also use (Ax^4 + B)/(Cx^4 + D). Do you know why I am using degrees 2,4,... though?
(Ax²+B)(x² - 9) / ((Cx² + D)(x² - 16))
Notice that we were careful to make our function have equal degree in the numerator and denominator (x^4 on the top and bottom).
This means that our horizontal asymptote is at y = A/C (leading coefficients are 1/1). We want this to be y = 2, so we have:
A/C = 2 *EQN 1*
Meanwhile, plugging in x = 0 gives us f(0) = 9B/16D. We want this to produce 1, so we set up:
9B/16D = 1 *EQN 2*
Can you come up with some examples of A, B, C, and D that satisfy *EQN 1* and *EQN 2*?
Can you think of a RULE for A,B,C,D to ensure that we don't get any extra vertical asymptotes? (HINT: Enforce that domain is still x ≠ ±4.)
Can you think of a RULE for A,B,C,D to ensure that we don't get any extra x-intercepts? (HINT: Enforce that the numerator still only produces two real roots.)
Can you think of a RULE for A,B,C,D that ensures both of the above?
Here is a Desmos graph (https://www.desmos.com/calculator/brkcqi3cbk).
