Using Graphs to Determine the Domain of a Function

4.12.4 Using Graphs to Determine the Domain of a Function

4.12.4 • Using Graphs to Determine the Domain of a Function

Activity

Consider the function $f (x) = \frac{6}{x - 2}$ .

To find out the sets of possible input and output values of the function, Clare created a table and evaluated $f$ at some values of $x$ . Along the way, she ran into some trouble.

1. Find $f (x)$ for each $x$ -value Clare listed. Describe what Clare’s trouble might be.

$x$	-10	0	$\frac{1}{2}$	2	8
$f (x)$

This table presents a set of x-values with an empty row designated for the corresponding f(x) function values.

Compare your answer:

When $x$ is $2$ , $f (x)$ is $\frac{6}{0}$ , which is undefined.

$x$	-10	0	$\frac{1}{2}$	2	8
$f (x)$	- $\frac{1}{2}$	-3	-4	$\frac{6}{0}$	1

Input and output values for a function f(x), including an undefined point.

2. Use graphing technology to graph function $f$ . What do you notice about the graph?

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Compare your answer:

Graph showing two curves: one in quadrants III and IV, and one in quadrants I and IV, both asymptotic to the x = 2 and x-axis.

The graph splits where the $x$ -value is around 2. See graph.

3. Use a calculator to compute the value you and Clare had trouble computing. What do you notice about the computation?

Compare your answer:

When I tried to find $f (2)$ with a calculator, it gave an error message.

4. How would you describe the domain of function $f$ ?

Compare your answer:

The domain includes all numbers except 2.

Are you ready for more?

Extending Your Thinking

Why do you think the graph of function $f$ looks the way it does? Why are there two parts that split at $x = 2$ , with one curving down as it approaches $x = 2$ from the left and the other curving up as it approaches $x = 2$ from the right?

Evaluate function $f$ at different $x$ -values that approach 2 but are not exactly 2, such as 1.8, 1.9, 1.95, 1.999, 2.2, 2.1, 2.05, 2.001, and so on. What do you notice about the values of $f (x)$ as the $x$ -values get closer and closer to 2?

Compare your answers:

When $x$ is less than 2, as it gets closer to 2 (from the left side), the denominator in $6 x - 2$ becomes a smaller and smaller negative value, so the quotient, $f (x)$ , gets larger and larger in the negative direction. It does so very quickly, which explains why the curve bends down rather dramatically.
When $x$ is greater than 2, as it approaches 2 (from the right side), the denominator in $6 x - 2$ becomes a smaller and smaller positive value, so $f (x)$ gets larger and larger in the positive direction. It does so very quickly as well, which explains the graph shooting up if we trace it from right to left.
The expression $6 x - 2$ undefined at $x = 2$ and we can see that the two pieces of the graph shoot off in opposite directions as $x$ approaches 2 from the left and from the right, which explains the gap at $x = 2$ .