Today we're going to talk about multiplying and dividing functions. Remember, a function is just a special type of relationship between two sets of data. So we're going to do some examples both multiplying and dividing functions.
So let's do some examples multiplying functions. I've got the function f of x equals to 2x and another function g of x equals 5x minus 8. So for the first example, I want to find f of 3 multiplied by g of negative 3.
So I'm going to first find the value of f of 3, find the value of g of negative 3, and then multiply those two values together. So to find f of 3, I'm going to substitute 3 in for my variable x. So this is going to be 2 times 3. And 2 times 3 will just give me 6.
To find g of negative 3, I'm going to substitute negative 3 in for my value for x. So this will become 5 times negative 3 minus 8. 5 times negative 3 will give me negative 15. And negative 15 minus 8 is going to give me negative 23.
So now that I found my two values, I can simply combine them by multiplying 6 times negative 23. And 6 times negative 23 is going to give me negative 138. So I found that f of 3 times g of negative 3 is equal to the negative 138.
So for my second example I want to find f of negative 1 times g of negative 1. So I could see find this in a similar way to my first example, by finding f of negative 1 finding g of negative 1, and then multiply those two values together. But because my input value is the same, I can simply find f of x times g of x, and then evaluate that expression for the input value of negative 1.
So to find f of x times g of x, I'm simply going to multiply my two expressions together. So f of x is 2x. I'm going to multiply that by 5x minus 8. I'm going to multiply these by distributing.
So 2x times 5x is going to give me 10x squared. And 2x times negative 8 is going to give me negative 16x. So I could also rewrite f of x times g of x as f times g of x. And I know that it's going to be equal to this expression.
So now that I found what f times g of x is, or f of x times g of x, I can find f of negative 1 times g of negative 1, or f times g of negative 1.
So I'll substitute negative 1 in for my x variables, and simplify. Negative 1 squared will give me positive 1. 10 times 1 will give me 10. 16 times negative 1 will give me negative 16. And 10 minus a negative 16 will give me a positive 26.
So I found that f times g of negative 1 is equal to 26. And I would also find that to f of negative 1 times g of negative 1 equals 26 if I were to find the value of f of negative 1, find the value of g of negative 1, and multiply them together, we would still find the value of 26.
So let's do some examples dividing functions. I've got the function f of x, which is equal to 2x squared plus 9 and the function g of x, which is equal to negative x squared. So for my first example, to find f of 0 divided by g of 1, I'm going to first find f of 0, then I'll find g of 1, and then I'll divide those two values.
So f of 0 is going to be equal to 2 times 0 squared plus 9. Simplifying this, 0 squared is 0, 2 times 0 is 0, and 0 plus 9 is 9. So f of 0 is 9.
To find g of 1, I'm going to substitute 1 in for my x variable. So this will give me negative 1 squared. 1 squared will just give me one, with a negative in front, which is to same as just negative 1. So I found that g of 1 is negative 1.
I can combine these two values by dividing 9 divided by negative 1 will give me negative 9. So I found that negative 9 is equal to f of 0 over, or divided by, g of 1.
For my second example, I've got-- I want to find f of negative 4 divided by g of negative 4. So here, because my input value is the same, I can simply find an expression for f of x divided by g of x, and then evaluate that expression for x is equal to negative 4.
So to find f of x over g of x, I'll take my expression for f of x over my expression for g of x. And I know that I can also write that as f over g of x. It's the same thing. Now to find f of negative 4 over g of negative 4, I know that's the same as f over g of negative 4.
So I'll simply substitute negative 4 in for my expression for my x variables. So this will be 2 times negative 4 squared plus 9 in my numerator, over negative 4 squared in my denominator. Simplifying this, I have negative 4 squared, which will give me 16. And here, negative 4 squared will give me another 16.
Simplify, 2 times 16 will give me 32. 32 plus 9 will give me 41. And 41 over, or divided by, negative 16 will give me approximately negative 2.56. So I found that f of negative 4 over g of negative 4, or f over g of negative 4, is approximately negative 2.56.
And it's important to note that since we're dividing by a function, we know that the denominator of a fraction cannot be equal to 0. The domain of this expression, or function, is going to be restricted to when the value of g of x is not equal to 0. Because again, you can't have 0 in the denominator of a fraction.
So let's go over our key points from today. To find the product or quotient of two or more functions, evaluate each function separately and combine the values for each function. For a given value a in the domain of f of x and g of x, f of a times g of a equals f times g of a.
And similarly, f of a divided by g or a equals f divided by g of a. So I hope that these key points and examples helped you understand a little bit more about multiplying and dividing functions. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.