A. Reciprocals of Linear Functions
Completion requirements
A. Reciprocals of Linear Functions
Recall that the reciprocal of a non-zero number, \(x\), is \(\frac{1}{x}\). Similarly,
the reciprocal of a function,
\(f(x)\), is \(\frac{1}{f(x)}\), for non-zero values of \(f(x)\).
In the Investigation, you saw that a function and its reciprocal are related. A table of values, and the graph of the linear function \(y = x\) and its reciprocal \(y= \frac{1}{x}\) are shown. Notice that any two corresponding, non-zero \(y\)-values are reciprocals. Verify this in the table and on the graph.
The reciprocal of \(1\) is \(1\) and the reciprocal of \(–1\) is \(–1\), so if the point \((x, 1)\) or \((x, –1)\) is on the graph of a function, the same point is also on the graph of the function’s reciprocal. These points are invariant points. As such, the invariant points of the reciprocal function can be determined where the lines \(y = 1\) and \(y = -1\) intersect the graphs of \(y = x\) and \(y = \frac{1}{x}\). On the graph, the invariant points are located where the two curves intersect, \((–1, –1)\) and \((1, 1)\).
In the table, and on the graph, when \(x = 0\), \(\frac{1}{x}\) is not defined, so there is no corresponding point on \(y = \frac{1}{x}\). Complete the following table to see what happens as \(x\) approaches \(0\).
As \(x\) approaches \(0\), \(\frac{1}{x}\) becomes an increasingly large positive or negative number. On the graph, this is represented by the nearly vertical portions of the curve near \(x = 0\). These curves will approach the vertical asymptote \(x = 0\), but will never intersect it. Similarly, as \(x\) gets very large, \(\frac{1}{x}\) approaches \(0\). This results in a horizontal asymptote at the line \(y = 0\).
To determine what happens to the graph of the function as it approaches the horizontal asymptote, complete the following table.
In the Investigation, you saw that a function and its reciprocal are related. A table of values, and the graph of the linear function \(y = x\) and its reciprocal \(y= \frac{1}{x}\) are shown. Notice that any two corresponding, non-zero \(y\)-values are reciprocals. Verify this in the table and on the graph.
|
\(x\)
|
\(y = x\)
|
\(y = \frac{1}{x}\)
|
| \(-4\) | \(-4\) | \(-\frac{1}{4}\) |
| \(-2\) | \(-2\) | \(-\frac{1}{2}\) |
| \(-1\) | \(-1\) | \(-1\) |
| \(-\frac{1}{2}\) | \(-\frac{1}{2}\) | \(-2\) |
|
\(-\frac{1}{4}\)
|
\(-\frac{1}{4}\) | \(-4\) |
| \(0\) | \(0\) |
undefined
|
| \(\frac{1}{4}\) | \(\frac{1}{4}\) | \(4\) |
| \(\frac{1}{2}\) | \(\frac{1}{2}\) | \(2\) |
| \(1\) | \(1\) | \(1\) |
| \(2\) | \(2\) | \(\frac{1}{2}\) |
| \(4\) | \(4\) | \(\frac{1}{4}\) |
The reciprocal of \(1\) is \(1\) and the reciprocal of \(–1\) is \(–1\), so if the point \((x, 1)\) or \((x, –1)\) is on the graph of a function, the same point is also on the graph of the function’s reciprocal. These points are invariant points. As such, the invariant points of the reciprocal function can be determined where the lines \(y = 1\) and \(y = -1\) intersect the graphs of \(y = x\) and \(y = \frac{1}{x}\). On the graph, the invariant points are located where the two curves intersect, \((–1, –1)\) and \((1, 1)\).
In the table, and on the graph, when \(x = 0\), \(\frac{1}{x}\) is not defined, so there is no corresponding point on \(y = \frac{1}{x}\). Complete the following table to see what happens as \(x\) approaches \(0\).
As \(x\) approaches \(0\), \(\frac{1}{x}\) becomes an increasingly large positive or negative number. On the graph, this is represented by the nearly vertical portions of the curve near \(x = 0\). These curves will approach the vertical asymptote \(x = 0\), but will never intersect it. Similarly, as \(x\) gets very large, \(\frac{1}{x}\) approaches \(0\). This results in a horizontal asymptote at the line \(y = 0\).
| \(x\) | \(\frac{1}{x}\) |
| \(-1\) | |
| \(-0.1\) | |
| \(-0.01\) | |
| \(-0.001\) | |
| \(0\) | undefined |
| \(0.001\) | |
| \(0.01\) | |
| \(0.1\) | |
| \(1\) |
To determine what happens to the graph of the function as it approaches the horizontal asymptote, complete the following table.
| \(x\) | \(\frac{1}{x}\) |
| \(-10 \thinspace 000\) | |
| \(-1 \thinspace 000\) | |
| \(-100\) | |
| \(-10\) | |
| \(10\) |
|
| \(100\) | |
| \(1\thinspace 000\) | |
| \(10 \thinspace 000\) |