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For any given rational function, differentiate between a function’s vertical and horizontal asymptotes. In two or more complete sentences, make a connection between the asymptotes and the function’s domain and range.

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MissPhiladelphia
For any given rational function, the vertical asymptotes represent the value of x that will make the denominator of the function equal to zero. The horizontal asymptote represent the value of y that results to an undefined value of x. The asymptotes serve as limits for the domain and range of the function.

Consider the function

f(x) = [tex]\frac{x-3}{x-4}[/tex]

Domain of the function = All real numbers except , x≠4 .

[tex]y=\frac{x-3}{x-4} \\\\ xy - 4y = x-3 \\\\ x y -x= 4 y-3\\\\ x=\frac{4 y-3}{y-1}[/tex]

Range = All real numbers except , y≠1 .

Horizontal Asymptote= Since the degree of numerator and denominator of rational function  is same , So Divide coefficient of x in numerator by divide coefficient of x in denominator.

So Horizontal Asymptote , is : y=1

To get vertical asymptote, put

Denominator =0

x-4=0

x=4 , is vertical asymptote.

Domain = All real numbers except vertical Asymptote

Range = All real numbers except Horizontal Asymptote



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