In this lesson, our instructor Vincent Selhorst-Jones explores Composite Functions in greater detail. Youll learn about arithmetic combinations, notation for composite functions, and how to use them. Functions operate like machines, even like multiple machines, but how? With visual interpretations and four worked-out examples, Vincent expertly explains composite functions.
Two (or more) functions can interact with each other through good old arithmetic: addition, subtraction, multiplication, and division. These sorts of interactions are called arithmetic combinations. Here are the four types:
Sum: (f+g)(x) = f(x) + g(x)
Difference: (f−g)(x) = f(x) − g(x)
Product: (fg)(x) = f(x) ·g(x)
Quotient: (f/g)(x) = f(x)/g(x) [and it must be that g(x) ≠ 0]
A much more interesting idea is to compose two functions. Instead of giving both functions the same input, we give the input to just one function. Then we take the first function's output, and plug that in to the second function. The second function
is acting on the first function. [If this idea is confusing, make sure to watch the video where we see some analogies.]
For function composition, we can use the notation of f °g. [Read as "f composed with g."] If f °g acts on x, we have (f°g ) (x). This means g acts on x first, then f acts on whatever results. [Notice how the functions act in order of closeness
to the original input.]
Another (much easier) way to see (f °g ) (x) is in the function notation format we're already used to:
(x) = f
(Recommendation: If you see the ° notation [such as f °g], rewrite it in the "normal" format [such as f ( g(x) )]. This normally makes it easier to understand and solve problems.)
Working with composite functions might seem intimidating at first, but it's really just about plugging in appropriately. Each function has its own "rule", so composing multiple functions just means using the rules in succession.
This idea of plugging in is shown beautifully by the notation f ( g(x) ). The function g acts on x, then f acts on the resulting g(x). Since we almost certainly know what g(x) looks like from the problem, we just use that as input for f.
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.