Question 1092988
<br>Part (a): Let f be of the form
{{{f(x) = (ax+b)/(x+c)}}}<br>
The horizontal asymptote is the ratio of the coefficients of the x term in the numerator and denominator.
{{{a/1 = -4}}}
So a = -4.<br>
The vertical asymptote is where the denominator is 0: at x = 3.
{{{3+c = 0}}}
So c = -3.<br>
f(1) = 0:
{{{(-4+b)/(1-3)= 0}}}
{{{-4+b = 0}}}<br>
So b = 4.<br>
The completed rational function is
{{{f(x) = (-4x+4)/(x-3)}}}<br>
<br>Part (b): Let f be of the form
{{{f(x) = (rx+s)/(2x+t)}}}<br>
The horizontal asymptote is the ratio of the coefficients of the x term in the numerator and denominator.
{{{r/2 = -4}}}
So r = -8.<br>
The vertical asymptote is where the denominator is 0: at x = 3.
{{{6+t = 0}}}
So t = -6.<br>
f(1) = 0:
{{{(-8+b)/(2-6)= 0}}}
{{{-8+b = 0}}}<br>
So b = 8.<br>
The completed rational function is
{{{f(x) = (-8x+8)/(2x-6)}}}<br>
Note that is in the form required for part (b).  However, in simplified form the function is<br>
{{{f(x) = (-4x+4)/(x-3)}}}<br>
which is the same function as in part (a).