Find a rational function that satisfies the given conditions:
Vertical asymptotes: x=1, x=-2
X-intercept: (-5,0)
Vertical asymptotes: x=1, x=-2
y = 1/((x-1)(x+2))
X-intercept: (-5,0)
y = (x+5)/((x-1)(x+2))
Vertical asymptotes: x=1, x=-2 ---> must have had 2 denominators of the form
(x-1)(x+2)
X-intercept: (-5,0) ---> the numerator must have contained (x+5)
a possible function:
y = (x+5)/( (x-1)(x+2) )
or we could put in some constants such as
y = 7(x+5)/(3(x-1)(x+2))
To find a rational function that satisfies the given conditions, we can start by considering the vertical asymptotes and x-intercept.
1) Vertical asymptote at x = 1:
For a vertical asymptote at x = 1, we need a factor of (x - 1) in the denominator of the rational function.
2) Vertical asymptote at x = -2:
For a vertical asymptote at x = -2, we need a factor of (x + 2) in the denominator of the rational function.
3) X-intercept at (-5, 0):
An x-intercept occurs when the function value is equal to zero. To have an x-intercept at x = -5, we need a factor of (x + 5) in the numerator of the rational function.
Combining these conditions, we can write the rational function as:
f(x) = (x + 5) / ((x - 1)(x + 2))
So, the rational function that satisfies the given conditions is f(x) = (x + 5) / ((x - 1)(x + 2)).
To find a rational function that satisfies the given conditions, we need to consider the properties of vertical asymptotes and x-intercepts.
A rational function can be written in the form f(x) = P(x) / Q(x), where P(x) and Q(x) are polynomials.
1. Vertical asymptotes: For a vertical asymptote at x = a, the denominator Q(x) must have a factor of (x - a) but not in the numerator P(x).
In this case, we have vertical asymptotes at x = 1 and x = -2. So, the denominator Q(x) should have factors of (x - 1) and (x + 2). A simple choice for Q(x) that satisfies this condition is Q(x) = (x - 1)(x + 2).
2. X-intercept: For an x-intercept at (x, 0), the numerator P(x) must have a factor of (x - x).
In this case, we are given that the x-intercept is (-5, 0). So, we need a factor of (x + 5) in the numerator P(x). A simple choice for P(x) that satisfies this condition is P(x) = (x + 5).
Combining these factors, we can write the rational function as:
f(x) = P(x) / Q(x) = (x + 5) / [(x - 1)(x + 2)].
Therefore, the rational function that satisfies the given conditions is f(x) = (x + 5) / [(x - 1)(x + 2)].