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Finding the Domain and Range of Functions

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Today we're going to talk about finding the domain and range of a function. So we're going to do some examples finding the domain and range of different functions, both by looking at its graph and algebraically. Let's talk about the domain and the range of a polynomial function. I have here a graph, an example of a polynomial function f of x.

We define the domain as the set of possible x values for the function and the range as the set of possible y values for the function. So if we're looking at this polynomial graph, we can see that the domain is all real numbers, because the polynomial is defined at any value of x. So we see that the graph is covering the values of negative x and positive x. The range of this function is also all real numbers, because at any particular value of x, it can correspond to any value of y, either from 0 to negative infinity or from 0 to positive infinity. So the range is also all real numbers.

Let's talk about the domain and range of a square root function. I've got the function f of x is equal to the square root of x minus 2 and its corresponding graph. So the domain of a square root function has some restrictions because a square root of a negative number is undefined.

So our domain is going to be restricted to x values that make the expression underneath our radical non-negative. So we can write an inequality to find an inequality to express our domain. So we want the expression underneath the radical x minus 2 to be greater than or equal to 0. We don't want it to be a negative number.

So we can solve this for x. And we find that x has to be greater than or equal to 2. So we can say the domain of our square root function is x is greater than or equal to 2 for this example.

The range is going to not ever be negative. We can see that by looking at our graph. Our graph starts at 0 on the y-axis and continues to approach positive infinity on the y-axis. Even though it's growing slowly, eventually it's approaching positive infinity. So we say our range of this function y is greater than or equal to 0.

So now let's look at finding the domain and range of an example of a rational function. So I've got this function f of x being equal to 3 over x plus 2 and its corresponding graph. And this is a rational function. We have a polynomial in the denominator. And because we have a polynomial in the denominator, this could equals 0 for certain values of x. And this is going to represent a domain restriction, because dividing by 0 or having a in the denominator of a fraction would leave a function undefined.

So to determine the values of x, that would make this denominator 0, we want to know when x plus 2 is equal to 0. And then that value will be excluded from our domain. So x plus 2 is equal to 0. We'll solve this for x by subtracting 2 on both sides. And we find that x is equal to negative 2.

So that means negative 2 is excluded from our domain. And so we can write our domain as all real x values not equal to negative 2. So any real x value not equal to negative 2 can be in our domain.

To find the range of our function, we can look at our graph. As y is increasing on this side of our graph, we see that it's approaching but never actually equaling 0. And as y is decreasing on this side of our graph, we again see that it's approaching but never actually equaling 0.

So we can see that 0 is being excluded from the range of our function. And so we can write the range as all real y values not equal to 0 And we also know that 0 is not in the range of our function because there is no x value here that will give us the y value or an overall value of our function equal to 0.

So finally, let's look at another graph with no domain restrictions. I've got another example of a rational function, f of x is equal to 3 over x squared plus 2x plus 2. I know that all solutions to this expression being equal to 0 are going to be a domain restriction. We don't want the denominator to be equal to zero.

So I can examine whether this expression x squared plus 2x plus 2-- I can look at when that is going to be equal to 0. And I would do that do that by solving a quadratic equation. However, I also know that if the discriminant, the value of b squared minus 4ac, if that is less than 0, then I know that this equation is going to have no real solution.

So b squared minus 4ac-- the values of a, b, and c are going to come from my quadratic, the coefficients and the constant. So my value for b is 2. My value for a is 1. There's no number in front of x squared. And my value for c is 2. So simplifying this, 2 squared is 4. 4 times 1 times 2 is 8. And 4 minus 8 is negative 4, which is less than 0.

So since we have seen that our discriminant is less than 0 or negative, we know that this quadratic equation has no real solution, which means again that my domain has no restrictions to the values of x. So we can conclude that the domain of this function is all real values of x. And we can see that from the graph, that it looks like it's covering all x values.

So let's go over our key points from today. In a function, the domain is the set of all possible input values of the function and the range is the set of all possible output values of the function. Functions with a square root have domain restrictions because the square root of a negative value is undefined. And functions with a fraction have domain restrictions because division by 0 leaves the function undefined.

So I hope that these key points and examples helped you understand a little bit more about finding the domain and range of a function. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.