Multiplying functions
How to multiply two functions together
You can multiply functions together to get the product of the functions. There are two ways to multiply functions.
You can either input the value for ???x??? into each function and then multiply the outputs together, or you can multiply the functions together and then input the value for ???x??? and simplify. We write the product of two functions ???f(x)??? and ???g(x)??? as
???(f\cdot g)(x)=f(x)\cdot g(x)???
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An example of multiplying functions and evaluating the product at a particular point
Example
Find ???fg(-4)??? if ???f(x)=x+2??? and ???g(x)=x-5???.
We need to find ???(fg)(-4)???, which we could rewrite as ???(f)(-4)\cdot g(-4)???. So we can plug ???x=-4??? into each function and then multiply the results together.
First let’s find ???f(-4)???.
???f(-4)=-4+2???
???f(-4)=-2???
Now let’s find ???g(-4)???.
???g(-4)=-4-5???
???g(-4)=-9???
Then the product ???(fg)(-4)??? is
???(fg)(-4)=f(-4)\cdot g(-4)???
???(fg)(-4)=-2\cdot-9???
???(fg)(-4)=18???
We also could have multiplied the functions together and then plugged in ???x=-4??? to find the answer.
???(fg)(x)=(x+2)(x-5)???
???(fg)(x)=x^2-5x+2x-10???
Simplify by combining like terms.
???(fg)(x)=x^2-3x-10???
Plug ???-4??? in for ???x???.
???(fg)(-4)=(-4)^2-3(-4)-10???
???(fg)(-4)=16+12-10???
???(fg)(-4)=16+2???
???(fg)(-4)=18???
Video example of how to find the product of functions
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Let's walk through another example where we find the product of functions
Example
Find ???(gh)(x)??? if ???g(x)=x+6??? and ???h(x)=x-8???.
We need to find ???(gh)(x)??? by multiplying the functions together. Our answer will be a new function instead of a single number since there’s no numerical value assigned to ???x???.
???(gh)(x)=(x+6)(x-8)???
???(gh)(x)=x^2-8x+6x-48???
Simplify by combining like terms.
???(gh)(x)=x^2-2x-48???
When you have multiple functions, you can use some simple rules to find their sum, difference, product, or quotient.
As you might guess, finding the product of functions is as simple multiplying the functions together. When you multiply two functions together, you'll get a third function as the result, and that third function will be the product of the two original functions.
For example, if you multiply f(x) and g(x), their product will be h(x)=fg(x), or h(x)=f(x)g(x). You can also evaluate the product at a particular point. So if you want to know the value of the product at x=2, you can plug x=2 into the product function h(x) to find h(2)=fg(2)=f(2)g(2).
Alternately, instead of first finding the product function, and then evaluating at x=2, you could first evaluate both f(x) and g(x) at x=2, and then multiply those results together to get the product h(2).
