Find an equation of a rational function f that satisfies the given conditions.

Question

vertical asymptotes: x = −2, x = 0 horizontal asymptote: y = 0 x-intercept: 1; f(2)=1

Answer 1

A rational function is one that has a numerator and a denominator.

 

Vertical asymptotes are values of x in which the function gets closer to, but never touches.  We can find the vertical asymptotes by setting the denominator equal to zero.

 

Since the vertical asymptote is x=-2 and x=0,  our denominator of the function will be

 

x(x + 2) = x² + 2x

 

 

Horizontal asymptotes are values of y in which the function gets closer to, but never touches.  Since the horizontal asymptote is y=0 , the degree of the numerator must be less than the degree of the denominator.  And since the x-intercept is x=1, our numerator will be

 

x - 1

 

 

So far, we can assume the function to be

 

f(x) = (x - 1) / (x² + 2x)

 

 

Next, we need to use the condition f(2)=1.   If we plug in x=2, f(x) should equal 1. But we need to multiply the right side of the function by some constant C.

 

1 = C(2 - 1) / (2² + 2(2))

 

 

Solve for C.

 

1 = C / 8

 

8 = C

 

 

So the function will be

 

f(x) = [8(x - 1)] / [(x² + 2x)]

 

f(x) = (8x - 8) / (x² + 2x)