Hi,
I just corrected the question, please see it again. It is not 4x , it is 4th root of x^2 -5x
i got [0,5] U [5,infinity) ??? or is it [0,5], the first answer is wrong. So basically I need to find a domain.
Thanks
OK Your answer makes much more sense in the context of that problem, but it is still wrong.
\(\displaystyle f(x) = \sqrt[4]{x^2 - 5x}.\)
Now, if we are dealing with real functions, f(x) is not defined where \(\displaystyle x^2 - 5x < 0.\) You clearly understood that.
So you asked yourself for what values of x is \(\displaystyle x^2 - 5x \ge 0.\) You were still doing fine up to here.
\(\displaystyle x^2 - 5x \ge 0 \implies x(x - 5) \ge 0.\) This step was fine, but now things get tricky.
Obviously \(\displaystyle x(x - 5) = 0\ if\ x = 0\ or\ x = 5.\) So 0 and 5 are in the domain.
\(\displaystyle x(x - 5) > 0\ if\ x < 0 > (x - 5)\ or\ x > 0 < (x - 5).\) You have
THREE cases to consider.
\(\displaystyle x < 0 \implies (x - 5) < 0 \implies x(x - 5) > 0.\) So values less than 0 are in the domain.
\(\displaystyle x > 5 \implies (x - 5) > 0 \implies x(x - 5) > 0.\) So values greater than 5 are in the domain.
What if 0 < x < 5?
\(\displaystyle 0 < x < 5 \implies - 5 < (x - 5) < 0 \implies x(x - 5) < 0.\) So values such that 0 < x < 5 are
NOT in the domain.
Domain is \(\displaystyle (- \infty,\ 0] \bigcup [5,\ \infty).\)
Clear now?
x^2−5x
