 Two important properties of any function are its domain and range. We define the following, the domain of a function f is the set of allowable input values, while the range of f is the set of possible output values. One important thing to notice here is the domain is what you're allowed to do, while the range corresponds to what you could do. So let's say I have f of x equals square root of x, so again pulling in our definition. So again the way to think about the domain is that it's what you can work with. So here x is our independent variable and we're taking the square root of it. Since we can't take the square root of a negative number, x must be non-negative. So in interval notation that means that x is going to be in the interval from zero included up to infinity. Meanwhile the range consists of our possibilities. What could we get as the principal square root of x? Since the principal square root of x is always non-negative, f of x must be non-negative, and so our range is also going to be from zero included all the way to infinity. For whatever reason, human beings seem to find it easier to work with what we can't do. We can't have a pack of cookies before dinner. So in something like this, if we want to find the domain, we might start out by asking ourselves what can't we use. Since this is a quotient, the denominator can't be equal to zero. So we might say that Since this is an inequality, to solve an inequality, we'll ignore the inequality and find the critical values. Solving the corresponding equality, giving us our critical values 3 plus or minus square root of 37, the whole mess over 2. And now, like a good math student and a good human being, we acknowledge the inequality. We want that expression not to be zero. So we could test a point in each of our regions and find our domain, which is a little bit messy, so we might put it in a different form. As long as x is not equal to 3 plus or minus square root of 37 over 2, the denominator isn't zero. So our domain, x not equal to 3 plus or minus root of 37 over 2. How about the domain and range of f of x equals 3x plus 7? So again, we might start by thinking about what we can't work with. Well, our function tells us to multiply x by 3 and then add 7. But we can do that to any real number, so our domain is all real numbers. Remember that to determine the range, we want to know what we can get as an output. And this is generally more difficult. One approach is to let our output variable be y and solve for the input variable x. One way of looking at this is that if we want to get an output y, this formula tells us the input x that we need. So if we want to get an output y equals 8, then we need to use an input that will be computed by our formula when y is equal to 8, so we'll substitute. And that tells us if x equals 1 third, we'll get a value of 8. Or if we want to get y equals negative 3, we need to use the input minus 10 thirds. Or if we wanted to get y equals horrible frightening mess, we need to use the input. Well, this really isn't any different from the other two cases. x will be horrible frightening input minus 7 over 3. And that's just a number we won't worry about what its exact value is. And the thing to recognize here is that since any value of y can be used in this formula, then every value of y can be an output of our function. And so the range is all real numbers. While we could try to solve for the range like we did, it's generally hard or impossible to do so. Instead, we can rely on a few theorems about arithmetic operations and the properties of arithmetic operations. Unfortunately, this means that there will be in general no procedure we can always use to find the range. Every problem of finding the range is different. So let's experience some math. So for our domain, what can we put into this formula? Well, we can substitute any value of x into the formula, which means that the domain is all real numbers. To find the range, well, if our output is y, then we have y equals x squared minus 4x minus 7. Now let's solve for y. So this looks a little bit messy. Let's try and solve this by completing the square. So I'll move that minus 7 over so it's not cluttering up our first two terms, x squared minus 4x. We'll complete the square, and we know that y plus 11 is x minus 2 quantity squared. Or since we'd like to say something about y, let's solve for it. So now let's think about this. Since x minus 2 is the square of a real number, then it must be greater than or equal to 0. Another way of looking at this is the smallest that x minus 2 squared can be is 0. So whatever y is, it's at least 0 minus 11. And so that tells us that y has to be greater than or equal to minus 11. And so our range is f of x greater than or equal to negative 11. How about an absolute value function? So our domain is, and that's because we can substitute any real number into the formula 3 times absolute value x minus 5 plus 7. We could try to find the range in the same way we did before. It's even better if you could make an argument based on an inequality. Since g of x is 3 times an absolute value, well again, we know that the least the absolute value can be is 0. So g of x is going to be greater than or equal to 7.