A linear rational function is a function of the form `f(x)=\frac{a\mathrm{x}+b}{c\mathrm{x}+d},a, b,c, d \in \mathbb{R}, d \neq 0`

Linear Rational Function
Graph of the function `f(x)=\frac{1}{x}`

The graph of a linear rational function is a hyperbola, and you have already seen its formula. However, there is one detail you need to be aware of. If `ad-bc=0`, then the graph of the function is a line with the formula `f(x)=\frac{a}{c}`.

Properties of the Function

  • The domain of definition includes all real numbers except one. The denominator must not be zero, so we know that `cx + d \neq 0`. By rearranging this inequality, we find that the domain of the function is: `D_f = \mathbb{R}\backslash\left\{-\frac{d}{c}\right\}`
  • The range is `H_f = \mathbb{R}\backslash\left\{-\frac{a}{c}\right\}`
  • The function is unbounded and has neither a maximum nor a minimum.
  • For `ad-bc>0`, the function is increasing over its entire domain.
  • For ad-bc<0, the function is decreasing over its entire domain.
  • The function has two asymptotes: `x=-\frac{d}{c}, y=\frac{a}{c}`

Graph Sketch

When tasked with sketching the graph of a function, you can calculate its behavior, but this is a long and often difficult process. However, if you recognize that it is a linear rational function, there is a handy method for sketching its graph. If you can transform the function into the form `f(x)=\frac{a}{x+m}+n; a, m, n \in \mathbb{R}`, you can easily sketch its graph.

Converting a linear rational function to this form is straightforward – you simply divide the function into two polynomials:

`\begin{aligned}(&ax+b):(cx+d)=\frac{a}{c}+\frac{\frac{cb-ad}{c}}{cx+d}=\frac{a}{c}+\frac{cb-ad}{c^2x+cd}\\&\underline{-\left(ax+\frac{ad}{c}\right)}\\&\ \ \ \ \ b-\frac{ad}{c}\end{aligned}`

However, memorizing such a formula is probably unnecessary; it's always easier to simply divide the polynomials.

Convert the function `f(x)=\frac{x+2}{x+4}` into the other form:

`\begin{aligned}(&x+2):(x+4)=1-\frac{2}{x+4}\\&\underline{-(x+4)}\\&\ \ -2\end{aligned}`

So, we have: `1-\frac{2}{x+4} = \frac{x+2}{x+4}`

You know how to convert it. Now for the actual graph sketch according to the formula `f(x)=\frac{a}{x+m}+n; a, m, n \in \mathbb{R}`:

If `a>0`, then the function is increasing. If a<0, the function is decreasing. The parameters `m, n` determine the shift along the x and y axes. It shifts by `-m` along the positive x axis and by `n` along the positive y axis.

For the function `f(x)=\frac{1}{x+2}+1`, it looks like this:

Linear Rational Function

Let's try it again. Determine the domain and range of the function `f(x)=\frac{x-2}{x+2}`

The domain and range are obtained simply by substituting into the formula.

  • `D_f = \mathbb{R}\backslash\left\{-\frac{d}{c}\right\}=\mathbb{R}\backslash\left\{-2\right\}`
  • `H_f = \mathbb{R}\backslash\left\{-\frac{a}{c}\right\}=\mathbb{R}\backslash\left\{1\right\}`

To sketch the graph, we need to adjust the function:

`\begin{aligned}(&x-2):(x+2)=1-\frac{4}{x+2}\\&\underline{-(x+2)}\\&\ \ -4\end{aligned}`

From this form, you can easily sketch the graph:

Linear Rational Function