**Video Transcript**

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Transcript
**Function Operations**

Zooming along on his flying carpet, Jaanav heads for home. Oh geez, he’s caught in a sandstorm, again. As he’s covered with sand. He has sand in his hair, sand in his ears, it’s everywhere!

Back at home, he thinks, there must be a solution for this annoying problem. Eureka, Jaanav has an idea! He'll attach a sand-proof glass dome to his flying carpet and he can advertise this amazing new product onTV.

### Expressing Costs in a Function

Of course, he wants to make lots and lots of money from this business venture, so he must figure out all the **costs** to manufacture and advertise the product. Plus, he must figure out the **sales price** of each item and and how many he must sell to make a healthy **profit**. Jaanav figures out that cost to make each carpet is 100 gold coins plus a one-time cost of 150 gold coins - to buy the loom used to weave the carpets. The total cost to produce the sand-proof glass dome is 50 gold coins per dome plus a one-time cost of 100 gold coins to buy the dome-building machinery. To figure out the total cost, let's write the costs of manufacturing as two different **functions**, and then add them together.

Let C represent the costs associated with the product, and let 'x' represent the number of units. We can write the costs of just the carpets that he will produce as C_carpet(x) = 100x + 150 and the costs of the sand-proof domes as C_dome(x) = 50x + 100. Remember, the costs depend on the number of items, 'x', that will be produced.

### Adding Functions

To **calculate the sum (f+g)(x)**, you just have to add f(x) and g(x). For the two given functions this means: (C_carpet + C_dome)(x) =C_carpet(x) + C_dome(x). Now to calculate the total cost of a flying carpet with a glass dome, Jaanav has to **add** the **two functions** together. C_totalcost(x) = C_carpet + C_dome(x). The sum of the two functions is equal to 100x + 150 + 50x + 100. **Combine the like terms**,

and then write the expression: 150x + 250.

### Subtracting Functions

Now that Jaanav knows the production costs, he needs to consider the cost of the televison advertisement, and then figure out the selling price for each unit. The ad will cost a one-time fee of 750 gold coins. Based on all the cost information, he decides to sell the units for 250 gold coins each. We can write this information as the function, R(x). “R” represents the receipts after the cost of the advertisement. So, R(x) = 250x – 750.

How much money will be left over after paying all of the costs? This amount is the profit, and it's equal to the receipts minus the total cost. We can write this as the function P(x): P(x) = (R – C_totalcost)(x). Which equals (250x – 750) – (150x + 250). To simplify, don’t forget to **distribute** the **negative** across both **terms** inside the second set of **parentheses**. We can write the **expression** as 250x – 150x -750 – 250, and then **combine the like terms**, giving us P(x) = 100x -1000.

### Break-Even / Covering the Cost

How many carpets does Jaanav need to sell just to **break-even**? Break-even is the amount of money Jaanav needs to earn just to **cover** his **fixed costs**. This is getting complicated! Maybe Jaanav should call his accountant? NO, we can help him. To **determine** the break even amount of this business venture we just need to **solve the equation** that models his profit. If we set P(x) = 0, then we can **isolate** 'x' and find out how many carpets Jaanav should sell.
He needs to sell 10 domed carpets just to break even. Well done.

### Multiplying Functions

But can we **multiply two functions**? To calculate the **product** of the functions (f times g)(x), simply multiply f(x) and g(x). Let's take a look at an example. f(x) equals 2x + 3 and g(x) equals 4x - 2. So to find the product (f times g)(x) we have to multiply the two terms 2x + 3 and 4x - 2. For example, using the **FOIL-method** we get: 8x² - 4x + 12x -6. We can simplify this expression by combining like terms. That's it.

### Dividing Functions

And how do we divide two functions? Let's take a look. Just like multiplying two functions, you have to **divide both terms** - '3x+5' and 'x-2' - to get a new function - (f/g)(x).
Oh, oh! The commercial is on the television! Let’s watch
*Do you get dust in your eyes when flying around on your magic carpet? Do you need protection from the elements? Then you need the OCD2000! The Oriental Carpet Dome will protect you from anything Mother Nature has to throw at you except maybe hurricanes and tornados...*

2 commentsThanks for liking (and commenting) on our videos!

Very informative for math AND advertising! Thanks!