ordered pairs mapping diagram (-2, -8) (-1, -4) (1, 4) (2, 8) x y table of values x y -2 -8 -1 -4 1 4 2 8 We can represent a function using ordered pairs, a mapping diagram showing every input mapped onto exactly one output, and a table of values.

domain {-2, -1, 1, 2} range {-8, -4, 4, 8} We can also represent a function using a graph. This is an example of a discrete graph showing 4 distinct points. The domain is the set of all x-values and we have 4 elements in this set. The range is the set of all y-values and we have 4 elements as well. Notice that in a discrete graph, we don’t connect the points nor do we draw a line through the points.

domain: R (-∞, ∞) range: Unlike the discrete graph with distinct points, we have a line that keeps going on both directions without any breaks or holes. This is a continuous graph. In a continuous graph the domain and range cover more than just a few points. Every point on the line make-up the domain and range. For our domain, we have the set of all real numbers. We can either use the symbol for all real or use interval notation from negative infinity to positive infinity. We use a parenthesis every time we use the infinity symbols because infinity is not a number that can be reached. The range will be also be the set of all real numbers

Using the same graph, is this a function Using the same graph, is this a function? When we’re dealing with graphs, we can use the vertical line test to check whether a relation is a function. So here’s our vertical line and the idea is this line should only pass through one point everywhere on the graph. If it passes through more than one point, then the relation is not a function. We just observed that the line passed though only one point everywhere on the graph, so this is a function.  

Is this graph a function Is this graph a function? The vertical line touched the graph just once throughout, so this is a function.

Not a function Is this graph a function? Our line intersects more than one point here at x=-3. This means that -3 has two outputs 2 and -2. Infact, there’s an infinite number of vertical lines that intersect the graph but we only need at least one intersection to prove that this relation is not a function.

Not a function Here we have a graph of a vertical line and as you can see, we have more than one point of intersection. Every point on the graph intersects the vertical line so this is not a function. Remember that every vertical line graph is a relation not a function.

Here’s a graph of a horizontal line Here’s a graph of a horizontal line. We only have one point of intersection everywhere, so this is a function.