Question

Help please. What are the domain and range of the following question?

Answer 1

\[\sqrt{\frac{ 2+\left( x^{2}-3x+1 \right)^{4} }{ \left( x^{2}+4x+3\right)^{2} }}\]

Answer 2

Domain: Two things to look out for here: 1) Denominator should not be zero and 2) The expression within the radical sign should not be negative.
1) Denominator should not be zero: Factor the denominator and find what values of x will make it zero. Exclude those values from the domain.
2) The expression within the radical sign should not be negative: \((x^2-3x+1)^4\) is always positive because of the even power 4. Similarly, \((x^2+4x+3)^2\) is always positive because of the even power 2. Therefore, the expression inside the radical sign is always positive no matter what the x value is. So this does not restrict the domain.

Answer 3

I get the domain, how about its range?

Answer 4

Are you allowed to use a graphing calculator?

Answer 5

Since the expression within the radical is always positive, the entire function is always positive. When x approaches -1 or -3, the denominator approaches zero and the function approaches +infinity. So the upper bound is +infinity. To find the lower bound, find the minima of the function by equating f'(x) = 0 and solving for x.

Answer 6

\[
f(x) = \sqrt{\frac{ 2+\left( x^{2}-3x+1 \right)^{4} }{ \left( x^{2}+4x+3\right)^{2} }} =
\frac{ \sqrt{2+\left( x^{2}-3x+1 \right)^{4} } }{ \left( x^{2}+4x+3\right) } \\
f'(x) = ? = 0
\]

Answer 7

If you are allowed to use a graphing calculator that will be the easier approach.
http://www.wolframalpha.com/input/?i=range+of+sqrt%28%282%2B%28x^2-3x%2B1%29^4%29%2F%28x^2%2B4x%2B3%29^2+%29

Answer 8

So, it is hard to write down the solution without using a graphing calculator. But, thanks for your helps..