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I am not sure how to approach this problem. I know that the range contains infinity from intuition and cannot contain negative numbers because of the square root. But I am not sure if it contains any number.

I have no idea for the domain. I know I can plug in numbers and since the degree is odd, it can’t go on forever but I am stuck.

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    $\begingroup$ Factor it as $\sqrt{x^4(x+3)^2}$ note that for $x=0$ the value is zero, and the polynomial $x^4(x+3)^2$ will take values arbitrarily large. Use the intermediate value theorem to show that it must take any non-negative value. $\endgroup$ – conditionalMethod Dec 6 '19 at 18:46
  • $\begingroup$ $\sqrt{t}$ has the restriction $t\in[0,\infty)$. So this boils down to solving an inequality. $\endgroup$ – Andrew Chin Dec 6 '19 at 18:46
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For $\sqrt{{}\cdot{}}$ to be defined, you need that its argument is $\geqslant 0$.

Now the argument in your case is $x^6+6x^5+9x^4=x^4(x+3)^2$, which is clearly always $\geqslant 0$, and makes sense for all $x\in\mathbb R$.

Thus the domain is all of $\mathbb R$.

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Range:

Your intuition is correct. Compare your function to $\sqrt{x^6}$ and note that since you function is larger at higher values, it must approach infinity also. It also contains zero as you can just plug in $0=x$. And your function does not have any holes so it will approach all values between.

Domain:

Factor out $x^4$ to get $\sqrt{x^4(x^2+6x+9)}$ note that both $x^4$ and $x^2+6x+9$ are both positive for all real numbers. (You can write the latter as $(x+3)^2$.)

So your domain contains all real numbers.

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A square root can only take positive arguments (= what's under the square root-sign should be positive or zero). Hence you could take the following steps (whenever you're facing a problem like this, Luke Collings already showed that the domain will be $\mathbb{R}$:

  1. Factor the polynomial $x^6 + 6x^5 + 9x^4$
  2. Find zeroes of this polynomial using its factorisation
  3. Determine the signs of the function in between the zeroes
  4. The domain consists of those intervals on which the polynomial is positive or zero.

You then find the range by considering the possible values obtained in the domain.

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