State the domain in interval notation

Question

J(x) = 6 − (9x − 9)^9/5 

Answer 1

Lets focus on the radical part.

 

⁵√(9x - 9)⁹ = ⁵√(9x - 9)⁵ * ⁵√(9x - 5)⁴

 

                 = (9x - 9) * ⁵√(9x - 9)⁴

 

 

If we look at the 5th-root factor, we can have a negative value under the 5th-root because a negative number raised to an odd power is negative.  Therefore, negative numbers are in the domain of a nth-root , where n is odd.

 

For example,   (-x)⁵ = -x * -x * -x * -x *-x = -x⁵

 

So your domain is all real numbers.

Answer 2

Alison,

 

You can plug in any real number for x and have the result J(x) be a real number. This is because 9x - 9 can be any real number and when you raise it to the 9th power or (9/5)th power (I am not sure which it is), you will also get a real number. This would NOT be true if the exponent were 9/2.

 

Whether the exponent is 9 or 9/5, the domain is (−∞,∞). If you have any questions about the notation, please let me know.