+618 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 $f(x).$
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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
