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What Are Rational Functions?

Functions with $x$ in the denominator are called rational functions. These are in fractional form and have asymptotes. Asymptotes are invisible lines in the coordinate system that the graph moves toward, but never meets. Rational functions can look like the graphs in the figure below.

Theory

Rational Function

A rational function is expressed in the following form:

f (x) = \frac{g (x)}{h (x)},

where $g (x)$ and $h (x)$ are polynomials.

Graphs of the rational functions f(x), g(x) and h(x) plotted together

You can see the graph of the following functions in the picture above

\begin{array}{l} f (x) = \frac{1}{x} g (x) = \frac{2 x}{x + 1} h (x) = \frac{x^{2} + 6 x - 3}{x^{5} + x^{3} - 1} \end{array}

\begin{array}{l} f (x) = \frac{1}{x} g (x) = \frac{2 x}{x + 1} h (x) = \frac{x^{2} + 6 x - 3}{x^{5} + x^{3} - 1} \end{array}

Hyperbolas are among the simplest rational functions. Hyperbolas are the relationship between two linear functions, $a x + b$ and $c x + d$ .

Theory

Hyperbolas

The hyperbola is an important rational function. It represents the relationship between two linear functions. The formula for a hyperbola is

f (x) = \frac{a x + b}{c x + d} .

Hyperbola with corresponding horizontal and vertical asymptotes in dotted lines

A hyperbola has a vertical and a horizontal asymptote (the dotted lines).

Formula

The Asymptotes of a Hyperbola

Vertical asymptotes

are found where the denominator is equal to zero. You can use this formula:

x = - \frac{d}{c}

Horizontal asymptote

is the value the graph goes towards when $x \to \pm \infty$ (read as, when x approaches positive or negative infinity). You can use this formula:

y = \frac{a}{c}

Example 1

Consider the following hyperbolic function:

f (x) = \frac{2 x + 2}{x - 1}

Find the asymptotes and the intersection between the graph and the axes.

The general formula for the hyperbola is

f (x) = \frac{a x + b}{c x + d}

In this exercise, $a = 2$ , $b = 2$ , $c = 1$ and $d = - 1$

You can find the vertical asymptote in this way:

x = - \frac{d}{c} = - \frac{(- 1)}{1} = 1 .

The vertical asymptote is $x = 1$ . Then you find the horizontal asymptote:

y = \frac{a}{c} = \frac{2}{1} = 2

The horizontal asymptote is $y = 2$ . The intersection with the $y$ -axis is found by inputting $x = 0$ , because the $x$ -coordinate is 0 along the entire $y$ -axis. This gives

f (0) = \frac{2 \cdot 0 + 2}{0 - 1} = \frac{2}{- 1} = - 2

The hyperbola intersects the $y$ -axis at $(0, - 2)$ . The intersection with the $x$ -axis is found by putting $f (x) = 0$ , because the $y$ -coordinate is 0 along the entire $x$ -axis. This gives

f (x) = \frac{2 x + 2}{x - 1} = 0

It is sufficient to set the numerator equal to zero, since a fraction is zero as long as the numerator is zero. Then you get

\begin{array}{l} 2 x + 2 & = 0 \\ 2 x & = - 2 | \div 2 \\ x & = - 1 \end{array}

The hyperbola intersects the $x$ -axis at $(- 1, 0)$ .

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