3. determine the equation of the horizontal asymptote for each function.\na) $f(x)=\\frac{x}{x + 5}$\nb)…

3. determine the equation of the horizontal asymptote for each function.\na) $f(x)=\\frac{x}{x + 5}$\nb) $g(x)=\\frac{5x}{6 - x}$\nc) $h(x)=-\\frac{x + 3}{x - 3}$\nd) $k(x)=-\\frac{3x - 2}{6 - 4x}$

3. determine the equation of the horizontal asymptote for each function.\na) $f(x)=\\frac{x}{x + 5}$\nb) $g(x)=\\frac{5x}{6 - x}$\nc) $h(x)=-\\frac{x + 3}{x - 3}$\nd) $k(x)=-\\frac{3x - 2}{6 - 4x}$

Answer

Explanation:

Step1: Recall the rule for horizontal asymptote of rational functions

For a rational function (y = \frac{f(x)}{g(x)}=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots + b_0}), if (n = m) (degree of numerator (f(x)) equals degree of denominator (g(x))), the horizontal asymptote is (y=\frac{a_n}{b_m}).

Step2: Analyze function (a)) (f(x)=\frac{x}{x + 5})

Here, (n = 1) (degree of (x) in numerator) and (m = 1) (degree of (x) in denominator). (a_n=1), (b_m = 1). So, (y=\frac{1}{1}=1)

Step3: Analyze function (b)) (g(x)=\frac{5x}{6 - x})

Rewrite (g(x)=\frac{5x}{-x + 6}). (n = 1), (m = 1), (a_n = 5), (b_m=-1). So, (y=\frac{5}{-1}=- 5)

Step4: Analyze function (c)) (h(x)=-\frac{x + 3}{x - 3})

(n = 1), (m = 1), (a_n=-1) (coefficient of (x) in (-(x + 3)=-x-3)), (b_m = 1). So, (y=\frac{-1}{1}=-1)

Step5: Analyze function (d)) (k(x)=-\frac{3x - 2}{6-4x})

Rewrite (k(x)=-\frac{3x - 2}{-4x + 6}). (n = 1), (m = 1), (a_n=-3) (coefficient of (x) in (-(3x - 2)=-3x + 2)), (b_m=-4). So, (y=\frac{-3}{-4}=\frac{3}{4})

Answer:

a) (y = 1) b) (y=-5) c) (y=-1) d) (y=\frac{3}{4})