The graph of a function is the visual representation of every input-output pair. It turns abstract formulas into pictures that reveal domain, range, behavior, and much more at a glance. This section explains how to construct and read graphs of functions.
Quick Reference
| Concept | Description |
|---|---|
| Graph of $f$ | Set of all points $(x, f(x))$ in the coordinate plane |
| Table of values | Selected $(x, y)$ pairs used to sketch the graph |
| Domain (from graph) | Projection of the graph onto the $x$-axis |
| Range (from graph) | Projection of the graph onto the $y$-axis |
| Smooth curve | Connect plotted points with a smooth, free-hand line |
What Is a Graph?
If we represent the independent variable by $x$ and the corresponding output by $y = f(x)$, each pair $(x, y)$ determines a point in the coordinate plane. Plotting all such points produces the graph of the function.
The graph of a function $f$ is the set of all ordered pairs
$\{(x, f(x)) \mid x \in \operatorname{Dom}(f)\}.$The graph gives a complete picture of the function's behavior: where it is positive or negative, where it rises or falls, what happens at the boundary of the domain, and more.
The graph of a function $f$ is shown below.
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From this graph we can read off the following facts:
- $f(1) = 0$: the graph crosses the $x$-axis at $x = 1$.
- $f(2)=1
- For $x > 1$: $f(x)$ is positive and increasing.
- For $0 < x < 1$: $f(x)$ is negative and decreases without bound as $x$ approaches $0$ from the right.
- The domain of $f$ is $(0, \infty)$.
How to Plot a Function by Hand
To plot the graph of $f$ over an interval, follow these steps:
- Construct a table of values by choosing several values of $x$ in the interval and computing $y = f(x)$ for each.
- Plot the corresponding $(x, y)$ points in the coordinate plane.
- Draw a smooth free-hand curve through the points.
Plot the graph of $f(x) = x^3/2$ for $x \in [-2, 2]$.
Solution
Step 1. Build a table using a few values:| $x$ | $-2$ | $-1.5$ | $-0.5$ | $0$ | $0.5$ | $1$ | $1.5$ | $2$ |
|---|---|---|---|---|---|---|---|---|
| $y = x^3/2$ | $-4$ | $-1.6875$ | $-0.0625$ | $0$ | $0.0625$ | $1$ | $1.6875$ | $4$ |
This is exactly how computer programs such as MATLAB and Python's Matplotlib plot functions: they compute many $(x, y)$ pairs and connect them with line segments. With enough points, the result looks like a smooth curve.
Reading Domain and Range from a Graph
- The domain of $f$ is the projection of the graph onto the $x$-axis: the set of all $x$-values for which the graph has a point.
- The range of $f$ is the projection of the graph onto the $y$-axis: the set of all $y$-values that appear on the graph.
To test whether a specific $x$-value is in the domain, draw a vertical line through that $x$. If it hits the graph, that $x$ is in the domain. To test whether a specific $y$-value is in the range, draw a horizontal line through that $y$. If it hits the graph, that $y$ is in the range.
