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 =

1 Answer
Hammer
Jun 23, 2018

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#

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