Compare/contrast similarities & differences in domains of the two radical functions.

Question

Compare and contrast the similarities and differences in the domains of the two following radical functions f(x) and g(x) below:

          ____

f(x)=√x-3

 

            ____

g(x)=³√x-3

Answer 1

f(x) is a square-root function.  g(x) is a cube-root function.

 

In a square-root function, the domain is limited because by concept, you cannot take the even-root of a negative number.  This is because if you take a value and square it, the result is always positive.  In a cube-root function, the domain is all real numbers.  By concept, you can take the odd-root of a number for any sign. This is because if you take any value of sign and raise it to an odd power, the result is either positive of negative, depending on the sign on the base number.

 

Here are some example:

 

(2)² = 4

(-2)² = 4

 

If you take a value of any sign and raise it to an even power, the result remains positive.

 

(2)³ = 8

(-2)³ = (-2)(-2)(-2) = -8

 

The sign of the results accompany the sign of the base when raising the value to an odd power.

 

 

For f(x) , set the inequality      x - 3 ≥ 0   , and solve for x.  The value of x is the domain of f(x).

For g(x), the domain is all real numbers.

 

 

In summary, the domain of an odd-root function is all real numbers.  Not so much for even-root functions.