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# What is the range of the function (3x+1)/(x+8)

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What is the range of the function (3x+1)/(x+8)

Oct 22, 2017

#1
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It will have a horizontal asymptote  at   y = 3

The range will be  ( - infinity, 3) U ( 3 , infinity )

Oct 22, 2017
#2
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What is the range of the function (3x+1)/(x+8)

I am not sure how CPhill looked at this question. This is what I'd do.

Chris may well have an easier way to look at it

x cannot equal -8 because the denominator cannot be equal to zero.

If I do the algebraic division here then

$$f(x)=\frac{3x+1}{x+8}=3+\frac{-23}{x+8}\\ Now\;\;\frac{-23}{x+8}\;\;\text{cannot equal 0 so } f(x)\ne 3\\ Also\\ y=\frac{-23}{x+8}+3\\ \text{is of the from }y=\frac{a}{x-h}+k \\ \text{so it must be a hyperbola with asymptotes at x=-8 and y=3.}$$

the range Chris gave is certainly correct, I just wanted to discuss the question more. :)

Oct 22, 2017
edited by Melody  Oct 22, 2017
#3
+95884
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We will have a vertical asymptote  ar x  =  -8

As the graph approaches x = -8 from the left , the graph will approach infinity

As the graph approaches x = -8 from the right , the graph will approach  -infinity

And there is a horizontal asymptote at y = 3  which is not crossed by the graph

So.....the range is   ( - infinity, 3) U ( 3 , infinity )

Oct 22, 2017