Throughout this unit, we've learned how to graph linear equations and linear inequalities. We're going to wrap this unit up by talking about functions. All the linear equations that we've encountered are actually functions and we're going to learn why. Functions take some sort of input like Bob, they do something mysterious to it, and then get an output like o. This mysterious part is the part we're trying to figure out. It's the description or rule that changes Bob into o. Here is the list of the names of some people. These are going to be my input and here is a list of outputs. I'm going to pair each person with an output by showing some errors. Bob goes to the letter o, Jill goes to the letter i, and Ralph goes to the letter a, and so on. So in this relation, what's the relationship between these inputs and these outputs? Finish out this sentence using the best choice.
Well Bob gets mapped to o, Jill to i, Ralph to a, and so on. Every person gets mapped to the vowel that's in their name, even Peter. So these outputs are all the vowels in the name of the person.
We can also think of these inputs as one set, and these outputs as another set. Our relation is a rule that describes the relationship between this set of inputs and this set of outputs. In this case, the relationship is the vowels found in each persons name. This relation though is actually very special, it's actually a function. A function is unique because the inputs always give the same outputs. For example, Bob will always give us a letter o. Functions may have two different inputs like Tim and Jill that both produce the same output, in this case Jill and Tim both give us i. This is okay. This would not be a function because we can never have two different outputs, a and o, for one input, in this case, Bob. For a function, each input must correspond with exactly one output. So there can only be one arrow leading from any input to an output.
Here I have one relation between age and person. The age are the inputs, and the type of person is the output. Here's a different relation: This one has an input of color, and this one has an output of fruit. I want you to tell me whether each relation is a function or not. Type yes or no in the boxes.
I know a 5 year old is considered to be a child, and so is a 2 year old. A person who is 16 would be considered a teenager, and people for these last ages would be considered adults. When checking to see if this is a function, I want to make sure that my inputs correspond to exactly 1 output. In each case, I only have one arrow leaving, or going away, towards the output. So yes, this is a function. And remember, just like last case with Jill and Tim, who's outputs were i, we have 5 and 2, who's outputs were both at child. We can havt two inputs that make the same output, but we can't have two outputs from one input. For the second relation I know the color red can correspond to a strawberry or to a raspberry. Notice this input has 2 different outputs. If I say the word red, I'm not sure what output I'm talking about. I could be talking about a strawberry or a raspberry. I don't know. So this is definitely not a function. It breaks our rule. This input has two outputs. We find the same case to be true with yellow. It could correspond to a banana or a lemon. If an input has two outputs even once, then it's not a function. So we only need to stop once we see that the first time.
So we know functions are a special relationship between a set of inputs and a set of outputs. This can work for objects and it can also work for numbers. Here's a set of numbers and here's another set of numbers. When we think of a coordinate point, we usually think of the x as the input and the y as the output. So these would be inputs, and these would be the corresponding outputs. Knowing that information, is this set of order pairs a function? And is this set of ordered pairs a function? Write yes or no in these boxes. As a hint, make a list of the inputs and the outputs, just like before, and then draw the arrows to map it out. Try your best.
Here my inputs are 4, 1, 4 and 5. So I list this here. I don't need to list this 4 twice so I'm going to erase this one. My outputs are 3, 2, 5 and negative is with 5, and 5 is with negative 1. Notice that I have an input leading to two outputs. So this first one is definitely not a function. We can see this more easily because the input repeats. If the input repeats, then that means it produces two different outputs which we know can't be true. So, if we have one x-value producing two different y-values, we know it's not a function. Here's a mapping diagram for the second set. The arrows indicate the ordered pairs, and I see that each input corresponds to exactly one output. So yes, this is a function.
Lets try using this concept of function to look at a graph. Remember at the beginning, I said that linear equations always represent functions, we know this because the input always produces exactly one output and we actually have a way of testing it. We can use a vertical line test to determine if a graph represents a set of points that is a function. If a vertical line only crosses the graph once then the graph represents a function. Just like in this case we know every x value on this graph has exactly one corresponding Y value. We know every input on this graph has exactly one corresponding output, or y value. So in this case we know the graph and its linear equation represent a function. Here's a completely different equation that you've never seen before. And this is its graph. If we use the vertical line test on this function. We can see that it hits the function once here, and then the vertical line crosses twice at two different places. This happens for many points along the graph. So we know this graph and this equation do not represent a function. Let's move over a quill line a little bit back and look at it when x equals 4. So here's our vertical line x equals 4 and I can see that I have two points on my graph. I have the point 4, 2 and 4, negative 2. Notice that the same input x equals 4 produces two different outputs, positive 2 or negative 2. Again this is why the graph is not a function.
Lets use this vertical line test and test some other graphs. Is this graph a function? Yes or no.
If I do the vertical line test, I can see that vertical lines only hit the graph at one point. So yes, this is a function.
How about this circle, is it a function?
If we do the vertical line test on the circle, this vertical line only intersects with the circle once, but I can see at other places the vertical lines hit twice. That means this input at x equals 1 has two different outputs. We could be up here or we could be down here. So no, this isn't a function.
How about this one? Do you think it's a function?
I can see that vertical lines would only intersect the graph once. So yes, this is a function. Every x value has one corresponding y value.
And finally, what about this one?
This one wouldn't be a function. Even though this vertical line only intersects the graph once, we have other vertical lines that intersect 2 or 3 times so definitely not a function.
For functions, we learn that they have a certain rule. We start with an input, something mysterious happens, and then, we get an output. This mysterious part represents the changes we make to go from the input to the output. For example, if I had an input of 1, I could get an output of 3. Something's happened here. You might have thought that I just added 2 to the number. Well, here's some other inputs. If I put in a 2, I get out a 5. And if I put in a 3, I get out a and outputs, what rule or expression could we write for this function? If I put in x, I would get out y. What are we doing to x? That's the rule I want you to write here.
I know if I multiply 1 by 2 and then add 1, I would get 3. Let's see if this same rule applies in the other cases. I can take my input of 2, multiply it by rule.
Because functions are so unique and have an input and output, we have another way of representing it. Instead of using y, we'll use something else. We'll use f of x, or a function of x because the output relies on the input x. F of x is the name of the function that represents the output. And x is our input variable. And this side of the equation describes how we change x. To find the value or output of a function, we use substitution. For example, we could evaluate this function for x equals 3. My input is 3, so I plug it in. I want to find the output, or f of 3, by carrying out this math. So, here, the output, when x is 3, is 7. When you see a statement like this, you really want to think of it in terms of the outputs and inputs. When the input x is equal to 3, the output of the function or the y value is 7. We don't have to evaluate our function for just numbers though. We could also use another expression. For example, if my input x was 3 x squared, I would plug it into my function. So, the output of 3 x squared would be 6 x squared plus 1. This would be the answer containing a variable. Notice, it's not just a number.
Here's a new function and I want you to try and evaluate it at these inputs. Put your final results in the boxes. Good luck.
For the first one, we plug in 3, in for x. So I have negative 2 times 3 plus And I wind up with 7. For f of negative 2, I plug in negative 2 for my x variable. That's my input. Negative 2 times negative 2 is positive 4 and these numbers sum to 11. For this last input, I have 4x squared, so I know my answer is going to contain a variable. I substitute 4x squared as my input and negative then positive 7 added to the end. Great job if you got all of those right.
Here is a different function and I've labelled it g of x. Functions don't have to always use the letter f. We often use the letter as f, g, and h, because it's the convention for algebra. So using this function, I want you to find these outputs g of 3, g of 0, g of negative 2, and g of 4. And be careful when squaring your input.
For this first problem, I plug in 3 for x, 3 squared is 9, and 5 times 3 is 15. Summing up these numbers I get 30. For g of zero, I let x equals zero, and that leaves me with 6. For g of negative 2, I substitute negative 2n for x. Negative these numbers up, I get zero. For g of 4, I substitute 4 in for x. 4 squared is