Graph Reciprocal Functions

(Video) Reciprocal Functions

by Textbook Tactics

Here's a video by Textbook Tactics explaining the reciprocal function.

Reciprocal Functions

Reciprocal functions are functions that are the multiplicative inverse of a function. We can mathematically demonstrate a reciprocal function by saying that the reciprocal of the function $f(x)$ is: $\frac{1}{f(x)}$. The reciprocal of a function is not the same as the inverse of a function. A function inverse is the reflection over the line $y=x$. To find the inverse of a function, you flip all of the $x$ and $y$ values.

For any given function, $f(x)$, the general form of the reciprocal function allowing for transformations is,

$$\frac{a}{x-h} + k$$

with the following key characteristics:

• Vertical asymptote at: $x=h$

• Horizontal asymptote at: $y=k$

• The denominator of a reciprocal function cannot be $0$

• Domain is all real numbers, except $0$

Let's look at a couple of examples.

First, we'll look at the classic, $y=x$. The reciprocal of this function is $y=\frac1x$. The following graphs demonstrate the original function and the reciprocal function.

The purple graph on the left is the function $y=x$. Its reciprocal function, $y=\frac{1}{x}$ is in green on the right. You'll notice that the green function does not touch the $x$-axis. This graph gets closer and closer to the $x$-axis, but never reaches it. The horizontal and vertical asymptotes of the green function are the axes.

Reciprocal Functions

Reciprocal functions are the multiplicative inverse of a function!

If we want to look for the reciprocal of a function, we can use the following formula,

$$\frac{a}{x-h} + k$$

, where $h$ considers a horizontal translation, $k$ considers a vertical translation, and $a$ considers a dilation (stretch or compression).

Rational functions can have discontinuities where the function is not defined.