The most basic rational function is [latex]f(x)=\dfrac{1}{x}[/latex]. In this section we will take a close look at the graph of this function, then later we will use it as a parent function and use transformations to create other rational functions.

Reciprocals

[latex]\dfrac{1}{x}[/latex] is the reciprocal of [latex]x[/latex]. So when [latex]x[/latex] is an integer (e.g., 4), it is quite simple to determine its reciprocal: [latex]\dfrac{1}{4}[/latex]. Likewise, the reciprocal of [latex]x=-3[/latex] is [latex]-\dfrac{1}{3}[/latex]. What about when [latex]x[/latex] is a fraction (e.g., [latex]\dfrac{1}{2}[/latex])? What is the value of [latex]\dfrac{1}{x}[/latex] (e.g., [latex]\dfrac{1}{\frac{1}{2}}[/latex])? What we are really asking is, what is [latex]1\div \dfrac{1}{2}[/latex]? Or, how many one-halves does it take to make 1? The answer is 2. There are two one-halves in 1 ([latex]2\times\frac{1}{2}=1[/latex]). Similarly, there are three thirds in 1([latex]3\times\frac{1}{3}=1[/latex]), there are four quarters in 1 ([latex]4\times\frac{1}{4}=1[/latex])… Therefore, we can conclude that

[latex]1\div\dfrac{1}{2}=1\times\dfrac{2}{1}=2[/latex]

[latex]1\div\dfrac{1}{3}=1\times\dfrac{3}{1}=3[/latex]

[latex]1\div\dfrac{1}{4}=1\times\dfrac{4}{1}=4[/latex]

[latex]1\div\dfrac{1}{x}=1 \times \dfrac{x}{1}=x[/latex]

1 divided by a simple fraction [latex]\dfrac{1}{x}[/latex] (i.e., [latex]1\div\dfrac{1}{x}[/latex]) is 1 multiplied by its reciprocal (i.e., [latex]\dfrac{x}{1}[/latex]).

[latex]\dfrac{1}{\frac{1}{x}}=1\div\dfrac{1}{x}=1\times\dfrac{x}{1}=x[/latex]

Now, what if the fraction is not a simple fraction (e.g., [latex]\dfrac{2}{3}[/latex])? The answer is still the same. It is 1 multiplied by the reciprocal of the fraction. That is,

[latex]\dfrac{1}{\frac{a}{b}}=1\div\dfrac{a}{b}=1\times\dfrac{b}{a}=\dfrac{b}{a}[/latex]

Consequently, the value of the function [latex]f(x)=\dfrac{1}{x}[/latex] when [latex]x=\dfrac{a}{b}[/latex] is [latex]f\left(\dfrac{a}{b}\right)=\dfrac{b}{a}[/latex].

Graphing the Rational Function [latex]f(x)=\dfrac{1}{x}[/latex]

We may create a table of values to sketch the graph of the parent rational function [latex]f(x)=\frac{1}{x}[/latex] (Table 1). We choose both positive and negative values for [latex]x[/latex] to obtain a spread of function values. In addition, we will choose fractional values of [latex]x[/latex] that lie between negative one and one as these fractions are significant to graphing the function. The smaller these fractions are, in absolute value, the larger the function value will be.

We can now plot the points and join the dots to create a smooth curve (Figure 1).

Figure 1. The graph of the function [latex]f(x)=\frac{1}{x}[/latex].

Features of Graph of [latex]f(x)=\dfrac{1}{x}[/latex]

There are three notable features for the graph of the function [latex]f(x)=\dfrac{1}{x}[/latex] (Figure 2). First the graph has two disconnected parts when [latex]x<0[/latex] and [latex]x>0[/latex]. In other words, the graph is not continuous as there is a gap at [latex]x=0[/latex]. This is very different from the other types of functions we have studied that are continuous. Such continuous graphs include linear functions, quadratic functions, polynomial functions, exponential functions, and logarithmic functions. The second feature is that the disconnected parts of the graph are separated by the [latex]x[/latex]-axis and [latex]y[/latex]-axis, which happen to be the horizontal and vertical asymptotes. The third feature is that the graph is symmetric. Let’s look at the features of symmetry and asymptotes in more detail.

Symmetry

A symmetric graph is a graph that is separated into two parts by the line of symmetry, where the two parts are mirror images of each other. If we were to fold the graph along the line of symmetry, the graph would fold onto itself. There are two perspectives for understanding the symmetry of the graph in Figure 1. One perspective is based on the line of symmetry [latex]y=x[/latex]. Under this line of symmetry, each part of the graph is symmetric with respect to the line [latex]y=x[/latex] (Figure 2). The other perspective is based on the line of symmetry [latex]y=-x[/latex]. Under this line of symmetry, the graph is symmetric with respect to the line of symmetry [latex]y=-x[/latex] (Figure 3).

Figure 2. Symmetry across the line [latex]y=x[/latex].

Figure 3. Symmetry across the line [latex]y=-x[/latex].

Asymptotes

A significant feature of the graph (Figure 2) of the function [latex]f(x)=\dfrac{1}{x}[/latex] is that the graph has two asymptotes. One is the horizontal asymptote [latex]y=0[/latex], i.e. the [latex]x[/latex]-axis, and the other one is the vertical asymptote [latex]x=0[/latex], i.e. the [latex]y[/latex]-axis. The graph gets closer and closer to the line [latex]y=0[/latex], i.e. the [latex]x[/latex]-axis, as the value of [latex]x[/latex] gets closer to positive infinity and negative infinity, but the graph never meets the [latex]x[/latex]-axis. Therefore the line [latex]y=0[/latex] is a horizontal asymptote of the function. The graph also gets closer and closer to the line [latex]x=0[/latex] as the [latex]y[/latex]-values get closer to positive infinity and negative infinity, but the graph never meets the [latex]y[/latex]-axis. Therefore, the line [latex]x=0[/latex] is a vertical asymptote of the function (Figure 4).

Figure 4. The asymptotes of the function[latex]f(x)=\frac{1}{x}[/latex].

If we lengthened our table of values (Table 1) we would discover why the graph never meets the [latex]x[/latex]-axis. As the value of [latex]x[/latex] gets larger and larger, the value of [latex]f(x)=\dfrac{1}{x}[/latex] gets smaller and smaller, i.e. closer to [latex]y=0[/latex]:

[latex]\begin{aligned}f(10)&=\frac{1}{10}=0.1\\\\f(100)&=\frac{1}{100}=0.01\\\\f(1000)&=\frac{1}{1000}=0.001\\\\f(10,000)&=\frac{1}{10,000}=0.0001\\\\f(10^n)&=\frac{1}{10^n}=0.000…0001&&(n-1)\text{ zeros after the point}\end{aligned}[/latex]

As the value of [latex]x[/latex] gets closer to positive infinity, the value of the function [latex]y[/latex] gets closer to zero. The value of [latex]y[/latex] will never be zero (i.e., meet the [latex]x[/latex]-axis) because the function value will always be a positive fraction when the value of [latex]x[/latex] is close to positive infinity. Similarly, as the value of [latex]x[/latex] gets closer to negative infinity, the value of the function [latex]y[/latex] gets closer to zero. The value of [latex]y[/latex] will never be zero (i.e., meet the [latex]x[/latex]-axis) because the function value will always be a negative fraction when the value of [latex]x[/latex] is close to negative infinity.

We can look at [latex]x[/latex]-values close to zero to see why the graph never meets the [latex]y[/latex]-axis:

[latex]\begin{aligned}f\left(\frac{1}{10}\right)&=10\\\\f\left(\frac{1}{100}\right)&=100\\\\f\left(\frac{1}{1000}\right)&=1000\\\\f\left(\frac{1}{10,000}\right)&=10,000\\\\f\left(\frac{1}{10^n}\right)&=10^n=1000…000&&n\text{ zeros after the }1\end{aligned}[/latex]

As the value of [latex]x[/latex] gets closer to zero from the positive side, the value of the function [latex]y[/latex] gets closer to positive infinity. The value of [latex]x[/latex] will never be zero (i.e., meet the [latex]y[/latex]-axis) because the function is not defined at [latex]x=0[/latex] (i.e., division by 0 is undefined). Therefore, the value of [latex]x[/latex] will always be a positive fraction that is close to zero, and the corresponding function value will get close to positive infinity. Similarly, as the the value of [latex]x[/latex] gets close to zero from the negative side, the value of the function [latex]y[/latex] gets closer to negative infinity. The value of [latex]x[/latex] will never be zero (i.e., meet the [latex]y[/latex]-axis) because the function is not defined at [latex]x=0[/latex]. Therefore, the value of [latex]x[/latex] will always be a negative fraction that is close to zero, and the corresponding function value will be close to negative infinity.

Domain and Range

The domain of the function [latex]f(x)=\frac{1}{x}[/latex] is all real numbers except the number zero because of the vertical asymptote [latex]x=0[/latex] (i.e., x cannot be zero). Therefore, the domain is {[latex]x | x \in \mathbb{R}, x\neq0[/latex]}. The range of the function [latex]f(x)=\frac{1}{x}[/latex] is all real numbers except the number zero because of the horizontal asymptote [latex]y=0[/latex] (i.e., [latex]y[/latex] will never be zero). Therefore, the range is {[latex]y | y \in \mathbb{R}, y\neq0[/latex]}.