Graphs of Functions
Contents
3. Graphs of Functions#
In this section, we’ll see how to graphically represent functions. We’ll also explore the domain and range of a function, given its graph.
3.1. Zoom In#
Let’s start with the most basic type of relation - a finite set of ordered pairs.
3.1.1. Graphs of Discrete Relations#
We’ve defined a relation as a set of ordered pairs \((x, y)\). For example, we can define a relation \(r = \{(1,3), (2, 1), (3, 5)\}\). When the inputs and outputs are real numbers (like in \(r\)), we can represent the same relation using a graph in the \(xy\)-plane, where \(x\) is represented by the horizontal axis and \(y\) is represented by the vertical axis.
Consider the point \((x, y) = (2, 1)\), which is pictured in blue. You can recover its \(x\)-value by squashing it onto the horizontal (\(x\)-)axis. You can recover its \(y\)-value by squashing it onto the vertical (\(y\)-)axis.
Practice
Based on the graph, try to identify the coordinates for the unlabeled points above. You can check your answer by clicking on each point. Investigate the graph above until you’re comfortable translating between a data point \((x, y)\) and its corresponding point on the \(xy\)-plane.
Answer
The graph depicts the relation \(\{(1,3), (2, 1), (3, 5)\}\) by plotting all three points.
3.1.2. Graphs of Functions Defined by Equations#
We can also graph functions that are defined by equations. Recall that an equation like \(y = x^2\) is interpreted as a function by thinking of \(x\) as the input and \(y\) as the corresponding output. The graph of this function consists of all \((x, y)\) that satisfy the equation.
The graph above is a graph of \(y = x^2\). It includes points like \((-1, 1)\) and \((2, 4)\), because those are \((x, y)\) pairs that satisfy the equation.
We could generate more points like this by choosing an arbitrary \(x\)-value and then plugging it into the equation to find its corresponding \(y\). For example, if you plug in \(x = 3\), you get \(y = 3^2 = 9\), so the point \((3, 9)\) would also be part of this graph.