Hey, everyone. So in the last video, we talked about adding and subtracting functions. Now, continuing on this theme of function operations, we're going to see how we can multiply and divide functions in this video. I will warn you that this process can be a bit tedious because when being asked to multiply or divide functions, you'll also likely be asked to find the domain of a resulting function that you get from this operation. And this process can be a little bit hard to keep track of, but don't worry about it because in this video we're going to be looking at some examples and scenarios that will hopefully clear up any confusion surrounding this topic. So let's get into it. We'll start by taking a look at multiplying functions. So if we have one function f of x which is the square root of x and we have g of x another function that's 3x minus 6, then multiplying these two functions together would just be the square root of x multiplied by 3x minus 6, and to simplify this I could take the square root of x and distribute it into these parentheses giving us that the product of our functions is 3xx-6x, And this right here would be our function when multiplying f and g together. Now we should also take a look at how the domain is going to behave because when multiplying functions, the domain is the set of numbers common to the domains of f and g. Now we talked about in previous videos how when we have the square root of x, the domain is going to be every positive number so in interval notation we're going to go from 0 to positive infinity. Now as for this other function g of x, notice we have this basic polynomial of 3x minus 6 and because of this, any number we plug in for x is going to be fine. There's nothing that's going to break our function. So our domain is simply negative infinity to positive infinity or basically just all real numbers. Now, the domain for f multiplied by g is going to be the combination of the domains individually for f and g. Since we see that our restriction for the first domain is 0 to infinity and that we have no restrictions for our second domain, then this combination is going to be 0 to infinity because since this is the only restriction we have, the total domain is going to look like this. But now let's take a look at what happens if we divide functions. So, in this case, we have the same two functions square root of x and 3x minus 6 and if we want to take f and divide it by g all we need to do is divide these two functions that we see. So we're just going to have the square root of x divided by 3x minus 6. Now, we already discussed before how the domain of the square root of x is 0 to infinity and the domain of 3x minus 6 is all real numbers. But if you take a look at what happened when we divided these two functions, you may notice there's actually another restriction that we have here. We have this denominator, and the denominator of a fraction can never be equal to 0. Since we said that this denominator here is g of x, that means the g of x cannot be 0, and the number that would make this denominator 0 is an x value of 2. So x cannot be equal to 2 because if you took 2 and replaced it with this x, you'd have 3 times 2 which is 6 and 6 minus 6 which is 0 and you cannot divide by 0. So the total domain that we have is going to be a combination of the restrictions we have for our 2 functions and the restriction we get from this combined function. So our total domain is going to go from 0 to 2 because we cannot have anything below 0 underneath the square root that we have, but we also cannot have anything that is equal to 2 because that's a restriction as well. So our domain is going to go from 0 to 2 and then we'll have another interval from 2 to positive infinity. So basically, we can have any real positive number that is not equal to 2. This is how you can find the domain after multiplying and dividing functions. Now before moving on to an example, there is one thing I want to mention is the notation may look different depending on what problem you have. So f multiplied by g this situation down here could also be written as f times g of x. So both of these notations mean the exact same thing and with division f of x over g of x could also be written as fg of x. So this is just something to keep in mind if you ever see this type of notation in problems. But now let's take a look at an example. So in this example, we are given these two functions. We're given f of x is equal to x squared minus 4 and g of x is x plus 2, and we're asked to complete the following operations below and find the domain of each function. So I'm first going to complete this operation which is f multiplied by g. So what I'm gonna do is take our function for f which is x squared minus 4 and then I'll take our function for g which is x plus 2 and I'll simply multiply these together. I can use the foil method here so I'll have x squared times x which will give us x cubed, x squared times 2 which will give us 2x squared, negative 4 times x which will give us minus 4x, and then I'll have negative 4 times 2 which will give us a minus 8. Now nothing here is going to combine any farther so this right here is the simplified expression. So this is basically our solution for f times g of x. Now to find the domain of this equation well, what I need to do is look at the domains for f and g and figure out how they're going to combine. But notice that we have 2 polynomials here, and for polynomials, all real numbers are going to be in the domain. So because of this, we could say that the total polynomial we got is going to have a domain which is simply all real numbers. So we'll just go from negative infinity to positive infinity. Nothing is going to break this function, so this would be the domain that we have. But now let's see what happens if we divide these two functions. So I'll take f of x which is x squared minus 4, and then I'll divide this by g of x which is x plus 2. Now I can simplify this fraction, but before doing that I'm actually going to find the domain first. We already talked about how these 2 polynomials that we initially plugged in are going to have a domain of all real numbers but notice that when we divide these 2 together we end up with a restriction because we have an X in the denominator. This denominator X plus 2 cannot be equal to 0, and if I go ahead and solve this little equation we have here I'll get the X cannot equal negative 2. So this is going to be a restriction for our domain. So now that I've found all the restrictions that we have here I'm gonna go ahead and simplify this. So x squared minus 4 is the same thing as x squared minus 2 squared because 2 squared is 4. And the reason I'm writing it like this is because this is a difference of two squares. We're taught that x squared minus 2 squared can also be written as x plus 2 times x minus 2, and then this is all divided by x plus 2. Now notice how the x plus twos are going to cancel in this equation, meaning all we're going to end up with is this right here which is x minus 2.
Table of contents
- 0. Review of Algebra4h 16m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 6m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 19m
- 10. Combinatorics & Probability1h 45m
3. Functions
Function Operations
Video duration:
7mPlay a video:
Related Videos
Related Practice
Function Operations practice set
