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A function f has a horizontal asymptote of y = -4, a vertical asymptote of x = 3, and a x-intercept at (1, 0).

 

Part A: Let f be of the form f(x) = (ax + b)/(x + c). Find an expression for f(x).

 

Part B: Let f be of the form f(x) = (rx + 3)/(2x + t). Find an expression for f(x).

 Sep 19, 2019
 #1
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Part A

 

The degree of the numerator  is equal to the degree of the denominator, so the horizontal asymptote is

y  =  a / 1     which we know is  -4

a / 1  = -4

a  =  -4

 

There is a vertical asymptote at  x =  3

So we know when  x = 3 ,    x + c  =  0

3 + c  =  0

c  =  -3

 

Now we know:     f(x)  =  (-4x + b)/(x - 3)

 

There is an x-intercept at  (1, 0)  so we know

f(1)  =  0

(-4(1) + b)/(1 - 3)  =  0

(-4 + b)/( -2 )  =  0

-4 + b  =  0

b  =  4

 

Altogether:     f(x)  =  (-4x + 4)/(x - 3)

 

Check:   https://www.desmos.com/calculator/ww1ibbxmyy

 Sep 19, 2019

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