Question

Answer the following questions for the function, r ( x ) = 3 x − 18/x^2+5 x+6 ? a)What is the domain of r ( x ) ? Give your answer using interval notation. b)r ( − 4 ) = c)For what value(s) of x does r ( x ) = 0 ? x= d)r ( x ) = − 3 5 . What is x ? x =

Answer 1

It is given

`r(x)=3x-18/x^2+5x+6=8x+6-18/x^2`

(a) The maximum domain of a function represents all the values it can take without becoming undefined/undetermined.

In our case, we see that

`r(0) = 6-18/0 -> " Undefined"`

As such, the domain of `r` is the real numbers without `0`.

`r:RR^"*"`

`RR^"*"=(-oo,+oo) "\"0=(-oo,0) uu (0,+oo)`

(b) `r(-4) = -32+6-18/16 =-217/8`

(c) We wish to find the roots of `r`:

`r(x)=0`
`8x+6-18/x^2=0`

This is going to be a bit tricky. First of all, multiply both sides by `x^2`, as `x` cannot be `0`:

`8x^3+6x^2-18=0`
`4x^3+3x^2-9=0`

See this link on how to solve this cubic equation.

Finally, we find out the only root of `r` to be

`x = 1/4 (-1 + root(3)(71 - 12 sqrt(35)) + root(3)(71 + 12 sqrt(35)))~~1.1022`

(d) `r(x) = -35`

`8x+6-18/x^2=-35`
`8x^3+6x^2-18=-35x^2`

I won't even try to solve this. By trial and error, you can approximate the roots of our new formed equation; for a fact, it has `3` real roots. I highly suggest you check out this Wikipedia page. All of the roots' exact forms contain complex numbers and are pretty "ugly".

Anyway, here are the three values of `x`:

`x_1 ~~-5.0363`
`x_2 ~~ -0.71422`
`x_3~~0.62552`