Linear fractional functions

Functions: Fractional functions

Linear fractional functions

Linear fractional function

A linear fractional function is a function of the form

$f(x)=\frac{\blue{a}x+\green{b}}{\purple{c}x+\orange{d}}$

where $\blue{a}$ , $\green{b}$ , $\purple{c}$ and $\orange{d}$ are numbers and $x$ is a variable.

The graph of a linear fractional function is a hyperbola with a vertical and horizontal asymptote.

plaatje

If $\purple c = 0$ then the graph of $f(x)$ is a line. We will often exclude this case.

The two branches of a linear fractional function always lie in opposing quadrants. They either lie in the first and third quadrant or in the second and fourth quadrant. We can even determine which of these is the case. It turns out that if $ad - bc < 0$ then the first case is true and if $ad - bc > 0$ then the second case is true.

Determine asymptotes linear fractional function

We determine the vertical asymptote of a linear fractional function $f(x)=\frac{\blue{a}x+\green{b}}{\purple{c}x+\orange{d}}$ by making the denominator $\purple{c}x+\orange{d}$ equal to $0$ and solving this equation.

Hence, for the vertical asymptote we find $x=-\frac{\orange{d}}{\purple{c}}$

The horizontal asymptote can be determined by noting that for very high values of $x$ the numbers $\green{b}$ and $\orange{d}$ are negligible relative to the terms with $x$ .

Hence, for the horizontal asymptote we have $y=\frac{\blue{a}x}{\purple{c}x}=\frac{\blue{a}}{\purple{c}}$

Take a look at the function $f(x)=\frac{\blue{2}x+\green{-3}}{\purple{4}x+\orange{2}}$

De verticale asymptoot is gelijk aan

$x=-\frac{\orange{2}}{\purple{4}}=-\frac{1}{2}$

De horizontale asymptoot is gelijk aan

$y=\frac{\blue{2}}{\purple{4}}=\frac{1}{2}$

Domain and range

Now that we know how to calculate te asymptotes, we also know the domain and range of a linear fractional function $f(x)=\frac{\blue{a}x+\green{b}}{\purple{c}x+\orange{d}}$ .

The domain is equal to all numbers except $-\frac{\orange{d}}{\purple{c}}$ .

The range is equal to all numbers except $\frac{\blue{a}}{\purple{c}}$ .

Below you see the graph of a function.

Of which of the following functions is this the graph?