How do you find the domain and range of #f(x)=-3x^(1/3)+4x^(1/2)#?

1 Answer
Jan 11, 2018

Domain is #{x:x!inRR,x>=0}#.
Range is #{f(x):f(x)!inRR,f(x)>=-0.25}#.


You find the domain by determining which numbers can be put in for x. In the case above, the square root of x is being taken. This means that x can never be less than 0, this is because the square root of a negative number can never be taken.

Hence, the domain is #{x:x!inRR,x>=0}#.

You find the range by determining which numbers can be a result of a certain x being put into f(x). Looking at the domain one can already see that only for values of x from 0-1 may the f(x) be less than 0. For all other values the square root of x will always be larger than the cube root of x, and therefore f(x) always larger than 0. So for x: 0-1 a graphing calculator can be used. Graph the function and look at the minima formed. The minima is found to be at -0.25.

Hence the range is #{f(x):f(x)!inRR,f(x)>=-0.25}#.