Finding the domain of a function

Question

Can someone explain the what the domain of a function is? How do I find the domain of the function f(x) = x^3

Answer 1

When you're finding the domain of the function, you have to find places where f(x) is undefined. If there are no points where f(x) = undefined, then the domain is all real numbers. There are only a few cases where the domain would be different: -log(x) has no points on or below zero, so to find the domain of log(stuff), then stuff > 0 -f(x)/g(x) (a rational equation) only has undefined points where g(x) = 0 because anything/0 is undefined. So you set the bottom ( g(x) ) not equal to zero and that's your domain -with square root equations, it can't have any points where x is less than zero, so you take the stuff under the square root and set it greater than or equal to 0. Hope this helped :D

Answer 2

The domain of a function is the set of all inputs (x values) such that the function returns a valid output (y values, or equivalently f(x) values).  Three things to watch out for in "state the domain" questions are:

(1) dividing by zero;  Because division by zero is undefined, if you see a fraction with an x in the denominator, set the denominator equal to zero and find out what x values make that happen (by solving your denominator equation for x).  Any x values making a denominator zero are not in the function's domain.

example:  f(x) = 5/(x-2)     in this function, x cannot equal 2. The domain is all real numbers not equal to 2.

 

(2) square roots of negative numbers (or more generally, "even" roots of negative numbers);  X values making a square root negative are not in the domain.

example:  g(x) = sqrt(2x+4)    in this function, 2x+4 cannot be negative.  Set up the inequality 2x + 4 < 0  and solve for x.  Get x < -2.  These x values are NOT in the domain of the function, so the domain is all real numbers greater than or equal to     -2.

 

(3) logarithms of non-positive numbers;  If you have a logarithmic function, make sure the part being logged is positive.

example:  h(x) = log(x+5)     in this function, x+5 must be positive.  To find out when it's negative (or zero) set up the inequality  x+5 <= (less than or equal to zero) 0.  Solve to get x <= -5.  These x values are NOT in the domain, so the domain is all real numbers greater than -5.

 

For your example, f(x) = x^3,  there are no fractions, no roots, and no logarithms.  The domain is all real numbers.