5. State a rational function that satisfies all of the following: The y – intercept is 24 There is a horizontal asymptote at y = 3 There are vertical asymptotes at x = -1 and x = 4
To find a rational function that satisfies the given conditions, we can start by considering the form of a general rational function:
ƒ(x) = (ax + b) / (cx + d)
Now, let's work through each given condition step by step and adjust the function accordingly.
1. The y-intercept is 24:
The y-intercept occurs when x = 0. Substituting x = 0 into the function, we get:
ƒ(0) = (a * 0 + b) / (c * 0 + d) = b / d = 24
This means that b = 24d.
So far, the rational function looks like this: ƒ(x) = (ax + 24d) / (cx + d).
2. There is a horizontal asymptote at y = 3:
Since there is a horizontal asymptote at y = 3, the ratio of the leading coefficients of the numerator and denominator of the function should be 3.
Therefore, a / c = 3.
Our rational function now looks like this: ƒ(x) = (3cx + 24d) / (cx + d).
3. There are vertical asymptotes at x = -1 and x = 4:
To have vertical asymptotes at x = -1 and x = 4, the denominator of the function should contain linear factors of (x + 1) and (x - 4).
The denominator can be written as: (cx + d)(x + 1)(x - 4).
So, now our rational function becomes: ƒ(x) = (3cx + 24d) / (cx + d)(x + 1)(x - 4).
This is the required rational function that satisfies all the given conditions.