Indicate domain of x using interval notation

Question

f(x)= 4sqrt(x^2-6x)

Answer 1

Okay Lucy, think about what's true about a square root function.  When can you not take a square root?  Negative numbers, right?  So your part under the radical must be > 0.

 

• x²-6x > 0   (the part under the radical)
• x(x-6) > 0  (factor)

 

When you have multiple zeros, they will split the number line into regions.  We need to test a point in each region:

 

1. Region 1, is x<0  If we plug in -1, this will give us a negative times a negative, so the expression would be positive
2. Region 2, 0<x<6, if we plug in 2, we get a positive times a negative, or a negative
3. Region 3, x>6, if we plug in 7, we'll get a positive times a positive, for a positive result

 

Now note that both regions 1 AND 3 bring back positive results.  Also remember that 0 is allowed.  So your domain would be:

 

(-infinity, 0] U [6, infinity)  (Note the 4 outside the square root does nothing to change the domain.)