**Introduction to Graphs of Functions**

It is useful to be able to draw pictures which represent functions. These pictures, or *graphs* , are a device for helping us to think about functions. In this book we will only graph functions whose domains and ranges are subsets of the real numbers.

We graph functions in the *x-y* plane. The elements of the domain of a function are thought of as points of the *x* -axis. The values of a function are measured on the *y* -axis. The graph of *f* associates to *x* the unique *y* value that the function *f* assigns to *x* . In other words, a point ( *x* , *y* ) lies on the graph of *f if and only if y* = *f* ( *x* ).

**Example 1**

Let *f* ( *x* ) = ( *x* ^{2} + 2)/( *x* − 1). Determine whether there are points of the graph of *f* corresponding to *x* = 3, 4, and 1.

**Solution 1**

The *y* value corresponding to *x* = 3 is *y* = *f* (3) = 11/2. Therefore the point (3, 11/2) lies on the graph of *f* . Similarly, *f* (4) = 6 so that (4, 6) lies on the graph. However, *f* is undefined at *x* = 1, so there is no point on the graph with *x* coordinate 1. The sketch in Fig. 1.38 was obtained by plotting several points.

**Math Note:** Notice that for each *x* in the domain of the function there is *one and only one* point on the graph—namely the unique point with *y* value equal to *f* ( *x* ). If *x* is not in the domain of *f* , then there is no point on the graph that corresponds to *x*.

**Example 2**

Is the curve in Fig. 1.39 the graph of a function?

**Solution 2**

Observe that, corresponding to *x* = 3, for instance, there are two *y* values on the curve. Therefore the curve cannot be the graph of a function.

**You Try It:** Graph the function *y* = *x* + ∣ *x* ∣.

**Example 3**

Is the curve in Fig. 1.40 the graph of a function?

**Fig. 1.40**

**Solution 3**

Notice that each *x* in the domain has just one *y* value corresponding to it. Thus, even though we cannot give a formula for the function, the curve is the graph of a function. The domain of this function is (−∞, 3) ∪ (5, 7).

**Math Note:** A nice, geometrical way to think about the condition that each *x* in the domain has corresponding to it precisely one *y* value is this:

If every vertical line drawn through a curve intersects that curve just once, then the curve is the graph of a function.

**You Try It:** Use the vertical line test to determine whether the locus *x* ^{2} + *y* ^{2} = 1 is the graph of a function.

Find practice problems and solutions for these concepts at: Calculus Basics Practice Test.

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