How to multiply functions and evaluate their product
When we multiply two functions, we use the same input in both and multiply their outputs. If we only need the product at one point, we can plug in first. Let’s do the same example both ways so you can see how the two methods fit together.
What does the product of two functions mean?
For each x where both functions are defined, the new function (fg) gives the product of their two outputs. In a formula, put parentheses around the expression for each function so you multiply all of its terms.
Example: Evaluate the functions first
Let f(x) = x + 2 and g(x) = x − 5. To find the product at −4, evaluate each function at that input. That gives f(−4) = −2 and g(−4) = −9.
Example: Multiply the functions first
We can use the same two functions to find a formula for the product at any x. Distribute each term in the first set of parentheses across both terms in the second, then combine the x-terms.
Both methods give 18. If we need to evaluate the product at several inputs, having the formula can save us a step each time.
Write the product as a new function
Now let g(x) = x + 6 and h(x) = x − 8. We’re finding the whole product function, so we leave x in the answer.
Find the domain of a product
The product can use every x that belongs to both domains. For example, if and g(x) = x, their product simplifies to 1. It still has the restriction x ≠ 0 because f(0) is undefined. With the two linear functions in our first example, every real x is allowed.
