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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 $f$

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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 $f(x).$

Sep 22, 2017

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f(x)   =  [ ax + b ] / [ x + c ]

a = -4      and  -3 + c  = 0 →   c = -3

And    if the x intercept   is (1, 0), then      -4(1)  + b  = 0   →  b  = 4

So

f(x) =   [  -4x + 4 ] /  [ x - 3 ]

See the graph, here :  https://www.desmos.com/calculator/u8akarzdzg

f(x) =  [ rx  +  s ]  / [ 2x + t ]

r =  -8           and       2(3) + t  = 0   →  t  = -6

And  if the x intercept   is (1, 0), then  -8(1)+ s = 0   → s  = 8

So

f (x)   = [ -8x + 8 ] / [ 2x - 6]

See the graph here : https://www.desmos.com/calculator/eeeod2ssah

Sep 22, 2017