Question

Determine the vertical intercept of h.h(0)=Determine the root(s) of h.x=Determine the vertical asymptote(s) of h.x=Determine the horizontal asymptote of h.y=

Answer 1

The general equation of line is :

`y-y_1=(y_2-y_1)/(x_2-x_1)(x-x_1)`

The equation of the function g(x) is:

Consider any two coordinates : (-3,0) & (0,3) So, the equation is:

`\begin{gathered} y-0=(3-0)/(0-(-3))(x-(-3)) \\ y=(3)/(3)(x+3) \\ y=x+3 \\ g(x)=x+3 \end{gathered}`

The equation of the function f(x) is:

Consider any two coordinates: (-1,0) & (0,1)

`\begin{gathered} y-0=(1-0)/(0-(-1))(x-(-1)) \\ y=1(x+1)_{} \\ y=x+1 \\ f(x)=x+1 \end{gathered}`

Since, h(x) = f(x)/g(x)

So, the funstion h(x) is express as:

`\begin{gathered} h(x)=(f(x))/(g(x)) \\ h(x)=(x+1)/(x+3) \end{gathered}`

a) Vertical intercept of h

Substitute x =0 in h(x)

`\begin{gathered} h(x)=(x+1)/(x+3) \\ h(0)=(0+1)/(0+3) \\ h(0)=(1)/(3) \end{gathered}`

b) Determine the roots of h(x)

`\begin{gathered} g(x)=(x+1)/(x+3) \\ \text{ the given expression is the iraationla polynomial } \\ (x+1)/(x+3)=(x+1)/(x+3) \\ \text{ Roots of h(x) =}(x+1)/(x+3) \end{gathered}`

c) Vertical asymptote : Vertical asymptotes are vertical lines which correspond to the zeroes of the denominator of a rational function

So,

`\begin{gathered} x+3=0 \\ x=-3 \end{gathered}`

The vertical assymptote : x =-3

D) Horizontal assymptote : x - 3

bAnswer: x -3

Answer:

a)h(0)=1/3

b) h(x)=(x+1)/(x+3)

c) x = -3

Hroizontal Assymotote : y =1/3