Hi, and welcome. My name is Anthony Varela. And today, we're going to evaluate functions. So we're going to look at a defined function f of x, and we're going to evaluate it at a certain value. We're also going to look at how we can use a graph of a function to evaluate it. And then we're going to talk about functions f of x and express f of x plus a, so x plus some other value, how we can express that function.
So first, let's talk about evaluating a function for a certain value. So if we're given the function f of x equals 2x squared minus 3, we can see in this notation that the argument of our function is x. So what is the argument of a function?
Well, the argument is the input value of the function, on which the value of the function depends. So if we saw 2z squared minus 3, we would say that as f of z, where z would be the argument. But we're seeing the variable x, so this is a function of x-- 2x squared minus 3. So the argument is what the function operates on, or it's that input value to the function.
Now, when we are evaluating f of a, we are substituting a for x in our function. So a is an actual value. We just substitute that in for the argument. So for example, if we'd like to evaluate f of negative 3, we're going to take our function f of x, but just write in negative 3 wherever we see x.
So f of negative 3 would be 2 times negative 3 squared minus 3. So negative 3 squared is a positive 9. Then we multiply that by 2 to get 18 and take away 3. We get 15. So f of negative 3 equals 15. That's just evaluating this function when x equals negative 3.
Let's do another one. Let's evaluate f of 2. So we're going to take our function f of x, but write in 2 every time we see x. So f of 2 equals 2 times 2 squared minus 3. 2 squared is 4. And then we multiply that by 2 to get 8, and then subtract 3. And we get that f of 2 equals 5.
So that's how we can evaluate a function at a given value. You just substitute that given value in for the argument, whatever that variable is. How can we use a graph to evaluate a function?
Well, this is the graph of y equals f of x. And on a graph, you just need to find x equals a, so whatever a value is given to you, and then find its corresponding y value. And that is going to be f of x-- or f of a I should say.
So let's evaluate f of 3 using this graph. So the first thing that we're going to do is find x equals 3. So we're taking that input value 3, finding it on the x-axis. And then what is the corresponding y value? And here it looks to be 6.
So y equals 6. Well, that means then that f of 3 equals 6. So just finding that corresponding y value on a graph is how we evaluate functions graphically.
Lastly, we're going to talk about how to express f of x plus a for a given a value. So simply, what we're going to do then is substitute x plus a in for x and then simplify our expression. So given the function 2x plus 7, or f of x equals 2x plus 7, I'd like to define f of x minus 2.
So we're going to start with f of x. So f of x equals 2x plus 7. But now, what we're going to do is every time we see x, we're actually going to write x minus 2. And keep them in parentheses. That's going to be an important step to make sure that you distribute everything that you need to and you simplify things correctly.
So now I have f of x minus 2. So just replacing x with x minus 2. And I have 2 times x minus 2 plus 7. So I'm just replacing x with x minus 2, making sure to group this in parentheses, because I have to multiply x minus 2 by 2 before we add 7.
So let's go ahead and distribute then the 2, so we get f of x minus 2 equals 2x minus 4. Then we have plus 7, so this simplifies to 2x plus 3. So f of x equals 2x plus 7, f of x minus 2 equals 2x plus 3. So we just substituted x plus a in for x and then simplified.
So let's review evaluating functions. First, we talked about the argument of a function. It's what the function operates on, so the input. So if we have f of x, this is a function of x. If we have g of r, that means that our argument would be r. The variable we would see would be r.
And if we have our function p of t, our argument is t. So our function might be t squared minus 4, for example. When evaluating a function, so f of a, you just substitute a for x in your expression for f of x. On a graph, you can find the a value on the x-axis and just find its corresponding y value. That would be of the value of the function at that point.
And lastly, we talked about expressing f of x plus a. So you would substitute an expression, x plus a into the function and simplify. So Thanks for watching this tutorial on evaluating functions. Hope to see you next time.
