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A function f has a horizontal asymptote of y = -4, a vertical asymptote of x = 3, and an 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+s)/(2x+t).Find an expression for f(x).

Guest Sep 24, 2018
 #1
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a)

In order to determine the vertical asymptote with a given rational function, you must set the denominator equal to zero and solve. Let's do that. 

 

\(x+c=0\) Set the denominator equal to zero and solve for x.
\(x=-c\) -c represents the expression that is the vertical asymptote. Since we know that the vertical asymptote is located at x=3, use the substitution property. 
\(3=-c\) Divide by -1 from both sides to solve for c.
\(c=-3\) Let's fill this information into the given rational function.
   

 

\(f(x)=\frac{ax+b}{x-3}\)

 

The horizontal asymptote has three conditions that you must always examine in any rational function. Every condition compares the degree of the numerator to the degree of the denominator. They are the following:

 

  • If the degree of the numerator is less than the degree of the denominator, then a horizontal asymptote occurs at y=0.
  • If the degree of the numerator is equal to the degree of the denominator, then a horizontal asymptote occurs at the ratio of the leading coefficients of the numerator and denominator.
  • If the degree of the numerator is greater than the degree of the denominator, then no horizontal asymptotes exist.

The degree of the numerator, in this case, is 1, and the degree of the denominator is also 1. According to the conditions above, we must find the "ratio of the leading coefficients of the numerator and denominator."

 

\(y=\frac{a}{1} \) The ratio of the leading coefficients are given on the left. We know that the horizontal asymptote of this particular rational function is at y=-4, so let's solve for by utilizing the Substitution Property of Equality.
\(-4=a\) We have determined the value of another variable, a
   

 

\(f(x)=\frac{-4x+b}{x-3}\)

 

Finally, we will use the last tidbit of information regarding this particular rational function, the location of the x-intercept. We only have one variable remaining, so we can just substitute this coordinate into the function and solve for b:

 

\(f(x)=\frac{-4x+b}{x-3}\) Substitute in the known coordinate of the x-intercept, (1,0).
\(f(1)=\frac{-4*1+b}{-1-3}\) Simplify the numerator and the denominator as much as possible. We know that f(1)=0, so substitute that into the function.
\(0=\frac{-4+b}{-4}\) Multiply by -4 on both sides. This ultimately cancels out the denominator.
\(0=-4+b\) Add 4 to both sides. 
\(b=4\)  
   

 

The final rational function, then, is \(f(x)=\frac{-4x+b}{x-3}\). You're done. 

 

b) 

 

This problem is almost exactly the same as the one I showcased above. Use the same techniques as I have showcased above, and you will be fine. 

TheXSquaredFactor  Sep 24, 2018

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