Let's continue our work with functions. There are basic operations we can perform with functions. We can add them, subtract them, multiply them, and divide them. We can even do something called the composition of functions, and we'll look at that later in this lesson. For now let's start with these two functions: f of x equals 3 x plus 2 and g of x equals x squared minus 1. Remember that x is the input for our function. It's what we can change, and input any number in here. F of x is the output for this function. It's the result that we get when we plug in x. And similarly, g of x is the output for this function. It's the result we get for whatever number we plug in for x. So, based on what you know about evaluating functions, what is f of 2? Write that here. Then what's g of 2? Put that one here. And then use what you know about f of 2 and g of 2 to figure out this question. What's f plus g of 2? Write that one here. Now, I know we haven't covered what this means yet, but I think you can figure it out.
F of 2 equals eight, g of 2 equals 3, and f plus g of 2 equals 11. Nice thinking if you got all 3 correct. To find the f of 2, we'll start by plugging the value of 2 in for X. So we'll have 3 times 2, which equals 6 plus 2, which equals 8. So when the input of the function x equals 2, the output of the function is 8. For g of 2 we'll use a similar approach. We'll plug in the value of 2 in for x, so we'll have g of two equals 2 squared minus 1. 2 squared equals 4, minus 1 is 3. So, we know that g of 2 is 3. And for this last one it just means we want to take the value of f of 2 and add it to the value of g of evaluated at x equals 2? Well, we know f of 2 equals 8 and we know g of 2 equals 3. So we're really just adding these two outputs together to get 11, and we can also see this on a graph. Here I've graphed f of x and g of x in Desmos. F of x is this straight orange line and g of x is this curved parabola. Notice that the f of 2, or the output for the orange function when x equals 2. Is positive 8 right here. If I zoom out, you can see that more clearly. And notice that the output for g of x, when x equals 2, is positive 3 here. So, we're just going to add the output here of 8 to the output here of 3. When we add the two outputs together, we'll get 11.
So, in terms of a graphical interpretation, we could just add the two y values here together to find f plus g of 2. We would just add 8 and 3 to get 11. So, what do you think would be f plus g of 0? You can either evaluate these functions for 0, or you can use the graph. When you think you have the answer, enter it here.
F plus g of 0 equals positive 1. Good thinking you found this number. We know that f of 0 equals 2, since when x equals 0, the function has a y value of 2. G of 0, however, is negative 1. When I plug in 0 in for this value of x for this function, we'll get a y value of negative 1. So, to find f plus g of 0, we simply added these two y values together. 2 plus negative 1, which equals positive 1.
But we don't have to evaluate functions just the numbers. We could find f plus g of x. This would be the function of adding these two functions together for any number x. In other words I want to create a function so that way when I plug in that number into it, I always get f plus g of that number. I want to have a function that can quickly add these two outputs together. So what do you think is this function? Type it in here.
F plus g of x equals x squared plus 3x plus 1. Great thinking if you found this function. To find this function, we simply want to add the two functions together. So, we'll have 3x plus 2 for f of x. And we'll have x squared minus 1 for g of x. So notice, here's f of x and here's g of x, we're just adding the two functions together. Write in the highest power first, of x squared plus 3 x plus 1. This is the result of adding these two functions together, f plus g of x. And let's see this on a graph as well. We have f of x, which is our straight line. We have g of x, which is our parabola. And then we have f of x plus g of x. It turns out that this is also a parabola. And remember before that we took f of 2, which that y value was 8, and we added it to g of 2, and that y value was 3. When we add the y values together here the value of 3 and here the value of 8, we get a new value of 11. Notice that the x coordinate for each of these points are all the same, they're all 2, but the y value for this point comes from adding these two y values together. So, this point came from adding f plus g of 2, but remember we also found another point, f plus g of 0. We found f plus g of 0 by adding the y coordinates for here together. We had 0, 2 on our orange graph, our f of x, And we had 0,-1 on our blue graph g(x). So, on our new graph f plus g of x, we have the value 0,1. Notice that the x coordinates are all the same and we get this y coordinate from adding these two y coordinates together. This is f of 0 and this is g of 0. Remember that if x equals 0, then the output for this function would just be 2. Since 3 times 0 equals 0, and we'll do that with positive 2. So we know this point is on the graph of f. Also, we know this point would be on the graph of g. When x equals simply add these two outputs together, and output of 2 and an output of negative 1, to get an output of positive 1. That's were we get this point from. And notice that this is actually represented for all values of x. Any input that we plug in here and here, we have their sum. This is another way we can think of their function using their graph.
We've just seen, adding functions together. So, what do you think about subtracting two functions? What do you think is, f minus g of 2, and what do you think is f minus g of x? This should just be one output, or one number and this should be a formula or function that involves X.
F minus g of 2 equals 5, and f minus g of x equals negative x squared plus 3 x plus 3. Great thinking if you found these two. To find f minus g of 2, we really just want to subtract the two outputs when the input for both functions is 2. So we just want to subtract the output of g when x equals 2 from the output of f, when x equals 2. We remember from before that f of 2 equals 8. This is the number we get when we plug x equals 2 here. And we can recall that g of 2 equals 3. This is the output for this function when x equals 2. So we really just want to subtract 3 from 8. So 8 minus 3 equals positive 5. And then to find this function we're going to take g of x and subtract it from f of x. So I've f of x first, which is 3 x plus 2, I'll write that here. Now I'm going to subtract this entire function g of x. That function is x squared minus plus 1. Remember that this negative sign will change the sign of both of these terms. Then we just add the like terms together to get this new function.
We've added and subtracted functions. So now lets try multiplying functions. What do you think would be the value of f times g of 2? Keep in mind that this is a multiplication symbol. We'll see a different symbol soon that means something else.
F times g of 2 equals 24. Great thinking if you found this number. When we find f times g of 2, we're really finding the product of these functions' outputs, when their input is x equals 2. We know f of 2 equals 8. And g of 2 equals 3. So we really just need to multiply these two outputs together. And what do you know, 8 times 3 equals 24, our answer.
How about the division of functions, what do you think would be f divided by g of 2? Write that answer, here.
Here the solution is 8 thirds, great thinking if you found this fraction. To find this number we'll have f of 2 divided by g of 2. We just want to take these two outputs when the functions inputs are x equals 2. We'll divide the output of g into the output of f. So f of 2 is 8 and g of 2 is 3, which is how we get 8 thirds. Now I want to caution you, this will only exist so long as this output does not equal zero. Remeber that fractions can not have zero in the denominator so, so long as the output of g of x does not equal zero then, f divided by g of some number will exist. In other words, we just want to make sure that this output for the function of g is not equal to zero.
Let's try some practice. Here are two new functions. And I want you to find each of these. Remember that for these you should have 1 number as the final output, and here you should have a function that uses x. Good luck.
Here are the solutions. Nice work if you got even half of these right. Now some of these might have been tricky, like the second one or even this last one. Let's walk through each of these to see how they're done. To find f plus g of take x and set it equal to 1 for this function. So, if we plug in a 1 here and here in for x, then we'll get an output of positive 1. Next we need to find g of 1. To find g of 1 we let x equal 1 for this function. So, we'll have 3 times to negative 2. So, we just add these two outputs together to get a final output of negative 1. To find the second one, f minus g of 1, we'll take the output from g of 1, negative 2, and we'll subtract that from the output f of 1, which equals 1. So notice I'll have 1 here minus negative 2. When we subtract a negative, we really add. So we'll have 1 plus 2 which equals positive 3. This is our output for f minus g of 1. For f times g of 1, we'll simply multiply these two outputs together. So, 1 times negative 2 equals negative 2. Now to find f divided by g of 2, we want to find f of 2 and divided it by g of 2. To find f of 2 we plug in 2 in for x. This would give us an output of 2. 2 squared equals 4, minus 4 is 0 plus 2 is 2. And for g of 2 we'll let the input x equal positive 2, which means we'll get an output of positive 1. We'll have 6 minus 5 which equals 1. Now we're ready to do our division, this is f of 2 and this is g of 2. We have 2 divided by 1 which equals 2. And finally to find f plus g of x, we'll simply add these two functions together. So, combining the like terms of x, we'll get one x and combining our constant terms we'll have negative 3. So, the overall function is x squared plus x minus 3, f plus g of x.
Nice work on functions up to this point. Now let's try this thing called the composition of functions. The composition of a function is when we use two or more functions together. The notation looks like this, f of g of x. Notice that this symbol is not a multiplication sign. It's a small open circle, that means the composite of g functions. It kind of looks like fog. But we really read this as f of g of x. To see what's happening here, let's look at some function machines. F of x will be a function that triples a number and g of x will be a function that adds 5 a number. So, we're going to start by finding the f of g of 3. What this means is, we want to find the output of g of 3 and use that as the input for f of x. In other words, we'll get some output here for plugging in 3 into g. Then we'll take this output, and plug it into f to get a new output. So, we know g of 3, we simply take 3 and plug it in for x. So g of 3 would equal 8. So, now just really want to find the f of 8. We take the output of g of 3, and plug that into f, so letting x equal 8 for this function, we'll have 3 times 8 which equals 24. This is f of g of 3. Knowing this, what do you think would be f of g of 5? Enter that number here.
F of g of 5 equals 30. Great problem solving if you found this answer. Remember when we find the f of g of 5, you really want to find g of 5 first. We want to find the output for g, when the input is 5. Then we take that output and we plug in that the input for f. So to find g of 5, we simply take 5 and plug it in here. So we'll have 5 plus 5, which we know is 10. So, the g of 5 equals 10. In other words, the output of the function g, when x equals 5, is 10. So now that we know this entire value, we can replace g of 5 with 10. Now we can just find f of 10. You want to find the output of the function f when x equals 10. I can remember that f was my tripling function. So, I'll have 3 times 10 or 30. So, we know the f of g of 5 equals 30. Our answer.
In the last problem, we found f of g of 5. For this problem, we're going to find g of f of 5. We're going to reverse the composition. Sometimes you'd want to do this, since functions might rely on each other in different ways. So what would be the output of the final result for this composition? Write that answer here.
This solution is positive 20. Great thinking if you found this number. To find g of f of 5, we first want to find the f of 5. We'll find the output for f when the input is 5, and then we'll take that output, and plug it in to g to get a new output. F was my tripling function, so I know the f of five would be 3 times 5 which equals 15. So we know the output of f when the input is 5 is 15. So we can replace this f of 5 with the number 15. So now we just need to find the g of 15 to get our answer. So when x equals 15, we'll plug that in here to get a new output. So plugging in 15 in for x here, we'll get 15 plus 5, which equals 20. That's how we know that g of f of 5 equals 20.
When we first look at the composition of two function, we first evaluated a function, add a number. And then we took that output and plugged it back into a new function to get a new output. But we wouldn't want to do this every single time for every single different number. Wouldn't it be nice if we just had one function that could do the composition for us? This is what we're going to find right now. F of g in terms of x. To find f of g of x, we're simply going to take the entire function, g of x and plug it in for the variable x. We'll plug this function into the other function as an input. So anywhere that we see x in our function for f, we're going to replace it with x plus 5. So notice that I have 3x here. So instead of 3x, I'll have 3 times, x plus 5, my new input. Notice how I took the entire function of g of x. And I put it inside as the input for f of x. Distributing this positive 3 will get the function 3x plus
What do you think would be the composite g of f of x? This would be the reverse composite of the one we just found. Write that answer here.
This function, g of f of x would be 3 x plus 5. Great thinking if you found this function. To find it, we take f of x, which is 3 x. And we plug it in anywhere inside of g where we see an x, here. So instead of having x plus 5. I'll have 3 x plus 5. Notice again, how f of x, this entire function became the input for g of x. There are no light terms here, so I simply drop these parethesis, to give myself the answer, 3 x plus 5.
For your first practice problem, try finding f of g of 1, f of g of 2, and f of g of x. These two should be numbers and this should be a function containing the variable x. Here's the function for f of x and here's the function for g of x. Good luck on this one.
Here are the solutions for 10, 1, and 9x squared minus 36x plus 37. Great solving, if you got all three of these correct. Now, if you only got one or two of these right, that's okay, too. These are not the easiest problems, and I'm sure you'll get it with more practice. To find f of g of 1, we first need to find the g of 1. So, when 1 is the input for g, we should get an output of negative 2. We get that simply by letting x equal positive 1. So, now, that we know g of 1 equals negative 2, we just need to find f of negative 2. We use negative 2 as the input for f. If we let x equal negative 2 for this function, here and here, then we can see that we'll get a total sum of 10. Negative 2 squared is positive 4. Negative 2 times negative 2 is positive 4, which makes use the same approach to tackle this problem, f of g of 2. Letting the x value equal 2 here, we get a g of 2 equal to 1. So we know that the output of g is 1 when x equals 2, so I can replace this g of 2 with the value 1. Now we just need to find f of 1. I'm letting the output of g when x equals 2, now be the input for f. Negative 2 and positive 2 sum to zero. So then I just have 1 squared, which equals positive 1. Now, we've done two calculations for f of g. But it would be so much nicer and easier if we just had one function to work with. An f of g of x function. So to find this function, we simply take g of x, and we plug it in anywhere we see an x and f. So notice that here and here, I've left spots for the x. We have something squared, minus 2 times something, plus 2. We're going to take g of x, which is 3x minus 5, and use that as our input. This is how we find a composition f of g of x. We have x as our variable, and we finding the output by plugging in g of x into this function. Squaring this binomial, we get 9x squared minus 30x plus 25. Then we'll distribute this negative 2 to get negative 6x plus 10, and a positive 2 on the end. There's no like terms with 9x squared, so I'll just list that here. Then we'll combine the like terms of negative 30x and negative 6x to get negative get our entire composite function.
Using the same two functions, this time I want you to find g of f of 1, and g of f of x. What are those two answers?
Here, g of f of 1 equals negative 2, and g of f of x equals 3x squared minus 6x plus 1. Great solving, if you got these two right. For this first problem, we'll start by finding the f of 1. We need to find the output, when the input for f is 1. So letting x equal 1 here, and here, we'll have a new expression. 1 squared equals 1, negative 2 times 1 is negaitive 2 and then we'll have positive 2 on the end. Summing these together we get a final output of positive with it's value, 1. Now we just need to find the g of 1. Plugging in 1 in for x here, will get 3 times 1 minus 5, so that equals negative 2. This is how we find g of f of 1. Now to find g of f of x we're going to take f of x this entire function and we're going to plug it in for this x here. Remember f of x is going to be the input for g. So wherever we see x for g we replace that with x squared minus two x plus 2. We distribute the positive 3 to get 3 x squared minus 6x plus 6. Then we just combine the like terms on the end, to get 3x squared minus 6x plus 1, g of f of x.
For this third practice problem, I've given you two new functions. Here's an f of x, and here's a g of x. What I want you to do is find g of f of 2, g of f of
Here the solutions are, negative 23, positive 5 and negative 4x squared minus one, we'll start by finding the f of 2. We'll take 2 and plug that in, for x in the function f. Doing so, we get an output of positive 12. So, when the input of f is 2, the output is 12. So, we just replace f of 2 with 12. Now we just need to find g of 12. We let g's input be 12, so negative 2 times 12 is negative 24, plus 1 is negative 23. This is how we get our first value. For the second problem, we want to find g of f of 0. So, we start by finding f of 0. You let x equals 0 for the function f, and then when we solve this, or simplify it, we get negative 2. So, we know f of 0 equals negative 2. So, we just replace this value with negative 2. So, letting x equal negative 2 for g, we get an output of positive 5. This is our second solution. And finally for g of f of x, we simply take the entire function f of x, and we plug it in as the input for g of x. So, wherever we see f of x, will replace that with 2x squared plus 3x minus 2, distribute the negative 2 will have negative 4x squared minus will have our final composite function.