Use interval notation to indicate the domain of F(x)

Question

f(x)= ⁴√(x^2-8x)

find the domain in interval notation 

Answer 1

Since nothing under an even root can be negative, we know that (x^2-8x) cannot be less than 0. So we simply find the solutions for x where (x^2-8x) is 0 or greater.

 

0=x^2-8x

0=x(x-8)

 

So x can be 0 or 8 for the function to equal 0. Now we need to test x for x<0, 0<x<8 and 8<x to see where x is positive.

 

We can use -1, 1, and 9 to test these intervals.

 

(-1)^2-8(-1)=1+8=9

Since 9 is positive, we know that the interval x<0 is an answer.

 

(1)^2-8(1)=1-8=-7

Since -7 is negative and nothing under an even root can be negative, we know that 0<x<8 is NOT part of the answer.

 

(9)^2-8(9)=81-72=9

Since 9 is positive, we know that the interval 8<x is part of the answer.

 

x<0 and 8<x are both parts of the answers, and x CAN equal either 0 or 8, so the interval notation for this looks like this:

(-∞,0] U [8,∞)

 

You use parentheses () when the number on that side is a limit, but the x CANNOT equal that number, or when the number is infinity because it is not a finite number. This is called being EXCLUSIVE.

 

You use brackets [] when the number on that side IS a possible value of x and is not infinity. This is called being INCLUSIVE.

 

The U means Union I believe, and means that BOTH of the intervals are possible answers at the same time.