In this lesson, our instructor Vincent Selhorst-Jones teaches the Idea of a Function and answers the question: What is a function? Vincent begins with a visual example of a function and gives a non-example. You will learn about function notation through some creative metaphors for functions. You will also learn how to use a function, how to use functions and tables, as well as, the relationship between input and output. The lesson ends with domain and range, and then five fully worked-out examples.
A function is a relation between two sets: a first set and a second set. For each element from the first set, the function assigns precisely one element in the second set.
Just like variables, it is useful to name functions with a symbol. Most often we will use f, but sometimes we'll use g, h, or whatever else makes sense.
If we want to talk about what f assigns to some input x, we show this with f(x). The first symbol is the "name" of the function, and the second symbol in parentheses is what the function is acting on.
There are many metaphors we can use to help us make sense of how a function works:
Transformation: The function transforms elements from one set into another. From problem to problem, the "rules" for transformation will (usually) change as we use different functions. However, as long as we're using the same function, the
rules never change: if we put in the same x, we will always get the same f(x) as our result.
Map: The function is a "map"-it tells us how to get from one set to another set. Of course, if we start at a different place, we might end up at a different destination. However, the map itself never changes: if we start at the same place,
we always arrive at the same destination.
Machine: The function is a machine that "eats" inputs and then produces outputs. What it produces depends on what it eats, but the machine is reliable: if it eats the same thing, it always produces the same output.
A very important idea is that, given the same input, a function will always produce the same output. If f(5) = 17 the first time, but then f(5) = 27 the second time, it is NOT a function. [However, there would be no problem with g(17) = 5 and
g(27) = 5.]
In math, most of the functions we'll work with (at least for the next few years) will take in real numbers and produce real numbers (ℝ→ ℝ).
When we are given a function, we will usually be told what its "rule" is: how it maps inputs to outputs. For example, if f(x) = 7x − 5, its rule is, "Multiply 7 and x together, then subtract by 5." [It's important to notice how x acts as a placeholder:
it tells us what happens to whatever the function acts upon.]
If we want to evaluate a function at a specific value, we just apply the "rule" to whatever our input value is. In practice, this turns out to be pretty simple: usually we are given a formula for each function, so we just follow the method of substitution.
Remember to wrap your substitution in parentheses, though! If you don't, you could make a mistake on a complex input.
f(x) = 7x − 5 ⇒ f(a−3) = 7(a−3) − 5
A good way to see the behavior of a function is by creating a table of values. On one side we have input values, while the other side shows us what the function outputs when given that input. Sometimes we'll be given the table, while other times
we will have to decide what inputs to use and determine the outputs ourselves.
The domain is the set of all inputs that the function can accept. While we generally assume that all of ℝ can be used as inputs, sometimes certain values will "break" our function: the output cannot be defined. Thus, our domain is all of ℝ except
that which breaks our function. [For now, we mostly have to watch out for dividing by zero and taking square roots of negative numbers. Later in the course we'll also have to be careful about inverse trigonometric functions and logarithms.]
The range is the set of all possible outputs a function can assign (given some domain). While these values will always be in ℝ (unless otherwise noted), they do not necessarily cover all of ℝ.
Idea of a Function
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.