Use the following information to construct a suitable rational function f(x).

* There is a vertical asymptote at x=2, and a horizontal asymptote at y=3.
* The graph of the function must go through the points (-4/3,0) and (0,-2)

To construct a rational function with the given information, we can follow these steps:

Step 1: Start by determining the equation for the horizontal asymptote.
Given that the horizontal asymptote is at y=3, the function must have a horizontal asymptote of y=3. Since horizontal asymptotes occur when the degree of the numerator is less than or equal to the degree of the denominator, the denominator of the rational function must have a greater degree than the numerator. Thus, the denominator of the function should be a polynomial of degree greater than the numerator.

Step 2: Write the equation of the vertical asymptote.
Given that there is a vertical asymptote at x=2, the denominator of the rational function must contain a binomial factor (x-2).

Step 3: Set up the general form of the rational function.
Let the rational function be f(x) = (ax + b) / (cx-2), where a, b, and c are constants to be determined.

Step 4: Use the given points to find the values of a and b.
Since the graph of the function must go through the points (-4/3,0) and (0,-2), we can substitute these values into the equation of the function.
For the point (-4/3,0): (a * (-4/3) + b) / (c * (-4/3) - 2) = 0
Simplifying, we have: (-4a + 3b) / (-4c - 6) = 0

For the point (0,-2): (a * 0 + b) / (c * 0 - 2) = -2
Simplifying, we have: b / (-2) = -2
Solving for b, we get: b = 4

Step 5: Substitute the value of b into the equation from Step 4.
(-4a + 3(4)) / (-4c - 6) = 0
-4a + 12 = 0
-4a = -12
a = 3

Step 6: Substitute the values of a and b back into the general form of the function.
f(x) = (3x + 4) / (cx - 2)

At this point, we have determined that the numerator of the rational function is 3x + 4. To find the value of c, we can use the information about the horizontal asymptote.

Step 7: Determine the value of c.
Since the horizontal asymptote is at y=3, the function should approach y=3 as x approaches infinity. This means that the value of c should make the denominator approach infinity when x approaches infinity.

To achieve this, we want to set the coefficient of x in the denominator (which is cx-2) to be positive. So, we choose c > 0.

Therefore, we can conclude that the rational function is f(x) = (3x + 4) / (cx - 2), where c > 0.

To construct a suitable rational function, we can start by considering the given information.

1. Vertical asymptote at x = 2: This means that the function cannot have any values at x = 2. To enforce this, we can include a factor in the denominator of the rational function that cancels out a factor in the numerator that would otherwise evaluate to zero at x = 2. A factor such as (x - 2) in the denominator would work.

2. Horizontal asymptote at y = 3: This indicates that as x approaches positive or negative infinity, the value of the function approaches 3. To ensure this, we need the degree of the numerator to be equal to or less than the degree of the denominator. Therefore, we can have a constant or linear term in the numerator and a quadratic or higher-degree term in the denominator.

3. The graph passes through (-4/3, 0) and (0, -2): We can use these points to determine the numerator of the rational function. Since the y-coordinate is determined by the numerator, we can write the numerator as -2(x + k), where k is a constant to be determined.

Combining these factors, we can represent the suitable rational function f(x) as:

f(x) = -2(x + k) / [(x - 2)(ax + b)],

where k, a, and b are constants to be determined.

To find the values of k, a, and b, we substitute the given points (-4/3, 0) and (0, -2) into the rational function:

For (-4/3, 0):

0 = -2((-4/3) + k) / [((-4/3) - 2)(a(-4/3) + b)]

Simplify this equation to find the value of k.

For (0, -2):

-2 = -2((0) + k) / [((0) - 2)(a(0) + b)]

Simplify this equation to find the values of a and b.

Once you have determined the values of k, a, and b, substitute them back into the expression for f(x) to obtain the final rational function.