I never knew “functions” was a course but it seems to be. Grade 11 or Grade 12 seems to be when functions is course that stumped many high school students. I’m not sure why? I think of a function as a rule or tool or machine that does a specific set of procedures to a given input that results in an output. If I think of a function as a machine, the machine requires certain inputs before it can do something and then spits something out the other end. Below are a few examples of functions.
Examples of functions
f(x) = x^2 \\ y = e^x \\ f(\theta) = \sin \theta \\ h(x) = -5x + 6 \\ f(x) = 2 \\ g(x) = 6x^3 + 2x^2 - 3x + 7 \\ y = x
In each of the functions above some value for x is put into the function or set of rules and a value f(x ) say, is spat out.
How do we define a function?
A function f : \mathbb{R} \longrightarrow \mathbb{R} \text{ defined by } y=f(x) is such that if y_1 = f(x) \text{ and } y_2 = f(x) then y_1 = y_2. What exactly does this mean? Let’s consider some diagrams.
From the picture above we have that functions f_1, f_2 \text{ and } f_3 are functions but f_4 is not a function. Notice that in the first three pictures, each red dot maps to at most one blue dot. However, in the fourth picture, the first red dot maps to two distinct blue dots. This is NOT what a function is supposed to do and hence, f_4 is not a function. We can also consider graphs of mappings and we can determine which sketches of mappings are indeed functions. Let’s consider the following graphs.
Vertical Line Test
The vertical line test is another way to determine whether a mapping is a function or not. We need to make a sketch of the mapping to use the vertical line test. Once we have drawn our mapping, the next step is to draw vertical lines. If we can draw a vertical line that cuts the sketch of the mapping more than once, then the mapping is NOT a function. Above we have a few examples. Notice that for the mappings in the sketches 1-4, no matter where you draw a vertical line, the red line in this case, it only crosses the sketch once. This means that all these mappings are functions. However, the sketches of mappings 5 and 6 notice that the red vertical line in each case, crosses the mapping twice. These two mappings are NOT functions.
If you want some more practice working with functions, take a look at our practice worksheets related to functions.
