Find an equation of a rational function that satisfies the given conditions.
Vertical asymptote: x = 4
Horizontal asymptote: y = 0
X-intercepts: none
Y-intercepts (0, -1/4)
Multiplicity of 1
Removable discontinuity: x = 1
End behavior: x --> infinity, f(x) --> -infinity; x --> -infinity, f(x) --> infinity.
I can graph this, but I am unsure how to turn this into a function. Please help? Thanks
You want f(±∞) = +∞
This is impossible, since there is a horizontal asymptote at y=0.
hole and vertical asymptote:
(x-1) / (x-4)(x-1)
This also gives y=0 as the vertical asymptote, since the denominator has higher degree than the numerator.
not sure what has the multiplicity of 1. That is usually reserved for the roots, but you say there are no x-intercepts.
Now f(0) = (-1)/(-4)(-1) = -1/4 as desired.
So, apart from the end behavior, this looks good. See
http://www.wolframalpha.com/input/?i=(x-1)+%2F+((x-4)(x-1))
Okay, thank you, Steve
To find an equation of a rational function that satisfies the given conditions, we need to consider the characteristics of the function and use the information provided.
Given the vertical asymptote x = 4, we know that the denominator of the rational function should contain a factor of (x - 4).
Given the horizontal asymptote y = 0, we know that the degree of the numerator should be less than or equal to the degree of the denominator. To ensure the horizontal asymptote at y=0 is not crossed in the long run, we can use a lower-degree polynomial for the numerator. Since we have a multiplicity of 1 and no x-intercept, we need to have a linear factor in the numerator. This linear factor should be (x - a) where a is some constant.
We know that the y-intercept is at (0, -1/4), so when x = 0, the function should equal -1/4. Therefore, we have another term in the numerator of -1/4.
To include the removable discontinuity at x = 1, we should have a common factor of (x - 1) in both the numerator and denominator. Since it's a removable discontinuity, this common factor will cancel out when simplifying the fraction.
Putting it all together, we can form the equation of the rational function:
f(x) = (-1/4(x - 1)) / (x - 4)
This function satisfies the given conditions. To verify, you can graph it and see if it matches the given criteria.