Question

Hello! Can you help with part A & B? Thank you!

Answer 1

we have the functions

`\begin{gathered} g(x)=-2x^2+13x+7 \\ h(x)=-x^2+4x+21 \end{gathered}`

Part A

Equate both equations

`-2x^2+13x+7=-x^2+4x+21`

Solve for x

`\begin{gathered} -2x^2+13x+7+x^2-4x-21=0 \\ -x^2+9x-14=0 \end{gathered}`

Solve the quadratic equation

using the formula

a=-1

b=9

c=-14

substitute

`x=(-9\pm√(9^2-4(-1)(-14)))/(2(-1))`
`x=(-9\pm5)/(-2)`

The values of are

x=2 and x=7

The answer Part A

The distances are x=2 units and x=7 units

Part B

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

so

`f(x)=(-2x^2+13x+7)/(-x^2+4x+21)`

Rewrite in factored form

`\begin{gathered} f(x)=(-2(x+(1)/(2))(x-7))/(-(x+3)(x-7)) \\ \\ f(x)=((2x+1))/((x+3)) \end{gathered}`

The given function has a discontinuity at x=7 (hole), a vertical asymptote at x=-3

and horizontal asymptote at y=2