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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 $$f(x) = \frac{ax+b}{x+c}.$$ Find an expression for f(x).

Part (b): Let f be of the form $$f(x) = \frac{rx+s}{2x+t}.$$ Find an expression for f(x).

I don't really know what to do. I formed the equations $$f(1)=\frac{x+b}{1+c}$$ and $$f(1)=\frac{x+s}{2+t}$$, respectively getting that $$\frac{x+b}{1+c}=0$$ and $$\frac{x+s}{2+t}=0$$. Then should I set them to equal each other? I don't really know if that would help because I'd still have a bunch of variables...am I even using the right approach? Also, I don't know if I should be utilizing the asymptotes to help me solve this.

Jun 14, 2020

#1
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(a) By definition of horizontal and vertical asymptotes:

$$\begin{cases} \displaystyle\lim_{x\to\infty} f(x) = -4\\ 3 + c = 0\\ \dfrac{a + b}{1 + c} = 0 \end{cases}$$

That means

$$\begin{cases} a = -4\\ c = -3\\ b = 4 \end{cases}$$

(a, b, c) = (-4, 4, -3).

(b) is similar to (a).

Jun 14, 2020
#2
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I don't really know calculus yet so I don't really understand this. Is there another way?

Jun 14, 2020
#3
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The definition of a horizontal asymptote is a horizontal line to which the graph becomes closer and closer as x becomes a very large or very small number, but never actually reaches it. So it's a y value that's undefined. A vertical asymptote is the same thing, but an undefined value for x.

a) Since the vertical asymptote is x=3, we can see that c = -3, since placing x=3 into $$\frac{ax+b}{x-3}$$ is undefined. (The denominator of a fraction is undefined for 0).

Let f(x) equal y. Now we can find a by solving this equation for x.

$$x=\frac{3y-b}{a-y}$$. Using the same thought process as above, we find that a = -4

Now we can use our values c = -3 and a = -4 and plug them in:

$$y=\frac{-4x+b}{x-3}$$

All we have to do is find b. We still have on piece of information we haven't used: the point (1, 0). Plug these x and y values in.

$$0=\frac{-4+b}{1-3}$$

Now we can solve for b and find that b = 4.

So putting everything together, we have our final answer: $$f(x)=\frac{-4x+4}{x-3}$$

You can do the same thing for part b.

Jun 14, 2020
#4
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Oh thanks, this explanation helps a lot more.

Jun 14, 2020
#5
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No problem! Glad it helped :)

thelizzybeth  Jun 14, 2020
#6
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Wait, why is it $$x=\frac{3y-b}{a-y}$$?

Guest Jun 15, 2020