Find a rational function that satisfies the given conditions. Answers may vary, but try to give the simplest answer possible:
Vertical asymptotes x = -4, x = 5;
x-intercept (-2, 0)
Try
f(x)=(x+2)/((x+4)(x-5))
the term x+2 will give a x-intercept at x=-2, the denominator gives vertical asymptotes at x=-4 and 5.
See http://i263.photobucket.com/albums/ii157/mathmate/rational.png
To find a rational function that satisfies the given conditions, we need the vertical asymptotes to be at x = -4 and x = 5.
A rational function with vertical asymptotes at x = -4 and x = 5 can be represented by the equation:
f(x) = (x + 4)(x - 5) / (x - 5)(x + 4)
To find the x-intercept, we substitute y = 0 into the equation:
0 = (x + 4)(x - 5) / (x - 5)(x + 4)
Since the denominator cannot be zero, we can cancel out the common factors:
0 = 1
This means we do not have any y-intercepts.
Therefore, a rational function that satisfies the given conditions is:
f(x) = (x + 4)(x - 5) / (x - 5)(x + 4)
To find a rational function that satisfies the given conditions, we need to consider the properties of rational functions.
1. Vertical asymptotes at x = -4 and x = 5:
A vertical asymptote occurs when the denominator of a rational function equals zero. Therefore, we need factors in the denominator that correspond to x + 4 and x - 5.
2. x-intercept at (-2, 0):
An x-intercept occurs when the numerator of a rational function equals zero. Therefore, we need a factor in the numerator that corresponds to x + 2.
Combining these conditions, we can write a general form for the rational function as:
f(x) = (x + 2) / [(x + 4)(x - 5)]
This is a rational function that satisfies the given conditions of having vertical asymptotes at x = -4 and x = 5, and an x-intercept at (-2, 0).