3.6 Rational functions  (Page 2/16)

End behavior of $f(x)=\frac{1}{x}$

As the values of $x$ approach infinity, the function values approach 0. As the values of $x$ approach negative infinity, the function values approach 0. See [link] . Symbolically, using arrow notation

$$\text{As }x\to \infty ,f(x)\to 0,\text{and as }x\to -\infty ,f(x)\to 0.$$

Based on this overall behavior and the graph, we can see that the function approaches 0 but never actually reaches 0; it seems to level off as the inputs become large. This behavior creates a horizontal asymptote , a horizontal line that the graph approaches as the input increases or decreases without bound. In this case, the graph is approaching the horizontal line $y=0.$ See [link] .

Using arrow notation

Use arrow notation to describe the end behavior and local behavior of the function graphed in [link] .

Notice that the graph is showing a vertical asymptote at $x=2,$ which tells us that the function is undefined at $x=2.$

$$\text{As }x\to 2^{-},f(x)\to -\infty ,\text{ and as }x\to 2^{+},f(x)\to \infty .$$

And as the inputs decrease without bound, the graph appears to be leveling off at output values of 4, indicating a horizontal asymptote at $y=4.$ As the inputs increase without bound, the graph levels off at 4.

$$\text{As }x\to \infty ,f(x)\to 4\text{ and as }x\to -\infty ,f(x)\to 4.$$

Using transformations to graph a rational function

Sketch a graph of the reciprocal function shifted two units to the left and up three units. Identify the horizontal and vertical asymptotes of the graph, if any.

Shifting the graph left 2 and up 3 would result in the function

$$f(x)=\frac{1}{x+2}+3$$

or equivalently, by giving the terms a common denominator,

$$f(x)=\frac{3x+7}{x+2}$$

The graph of the shifted function is displayed in [link] .

Notice that this function is undefined at $x=-2,$ and the graph also is showing a vertical asymptote at $x=-2.$

$$\text{As }x\to -2^{-}, f(x)\to -\infty ,\text{and as} x\to -2^{+}, f(x)\to \infty .$$

As the inputs increase and decrease without bound, the graph appears to be leveling off at output values of 3, indicating a horizontal asymptote at $y=3.$

$$\text{As }x\to \pm \infty , f(x)\to 3.$$

Try It

Sketch the graph, and find the horizontal and vertical asymptotes of the reciprocal squared function that has been shifted right 3 units and down 4 units.

The function and the asymptotes are shifted 3 units right and 4 units down. As $x\to 3,f(x)\to \infty ,$ and as $x\to \pm \infty ,f(x)\to -4.$

The function is $f(x)=\frac{1}{{(x-3)}^2}-4.$

Solving applied problems involving rational functions

In [link] , we shifted a toolkit function in a way that resulted in the function $f(x)=\frac{3x+7}{x+2}.$ This is an example of a rational function. A rational function is a function that can be written as the quotient of two polynomial functions. Many real-world problems require us to find the ratio of two polynomial functions. Problems involving rates and concentrations often involve rational functions.