Help with logarithm domains please?

Question

What is the domain of:

f(x)= log₂(x³-X)

 

I know how to find the domain of logarithms, but I can't figure out how to treat the x³-x>0. Help please?

Answer 1

x³ - x > 0

x(x² - 1) > 0

x(x + 1)(x - 1) > 0

 

Now determine the domains for these possibilities:

1. All factors are greater than zero.

2. First and second are negative and third is positive

3. First and third are negative and second is positive

4. Second and third are negative and first is positive

 

For example: all factors are greater than zero.

 

x > 0   and   x + 1 > 0   and   x - 1 >0

x > 0   and   x > -1        and   x > 1

The domain is x > 1

 

Do 2., 3., and 4. the same way.