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help plz

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A function f has a horizontal asymptote of y=-4, a veritcal 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)

Jul 30, 2020
edited by unknowledgeable  Jul 30, 2020

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Since there is a vertical asymptote of x = 3, there will be a factor of (x - 3) in the denominator.

This makes c = -3.

Since there is a horizontal asymptote of y = - 4, the ratio ax/x must be -4.

This makes a = -4.

Since there is an x-intercept at (1,0), we can look at  y = (-4x + b) / (x - 3)

and substitute 1 for x and 0 for y:  0  =  (-4·1 + b) / (0 - 3).

This makes b = 4.

f(x)  =  (-4x + 4) / (x - 3)

To get the answer to b, multiply both the numerator and denominator by 2.

Jul 30, 2020