Let’s find the domain of a function. We got f(x) equals the quantity 1 plus root x times the quantity 4 minus root 9 minus x. For this function to be defined, we need both of these radicals to be defined. And this will be defined if x is greater than or equal to 0. This one will be defined if 9 minus x is greater than or equal to 0. And we need both of those to be true.
What does this mean? I can subtract 9 from both sides and get minus x is greater than or equal minus 9. Then I can multiply by -1. And of course doing so reverses the direction of the inequality. I get x is less or equal to 9 and I’ll reverse this; 0 is less than or equal to x.
Whenever you have 2 inequalities like this, they can be combined into a compound inequality. 0 is less than or equal to x is less than or equal to 9. And that means the domain of this function is the interval from 0 to 9.