# How to Use a Reciprocal Function

## What is a Reciprocal Function

A **reciprocal function** is the mathematical inverse of a function. In math, reciprocal simply means one divided by a number. So a reciprocal function is one divided by the function.

- The reciprocal of {eq}5 {/eq} is {eq}\frac{1}{5} {/eq}

- The reciprocal of the function {eq}x+5 {/eq} is {eq}\frac{1}{x + 5} {/eq}

The reciprocal function is the multiplicative inverse of the function. In other words, it is the function turned up-side down. Any function can be thought of as a fraction:

- {eq}x+5 {/eq} as a fraction is {eq}\frac{x+5}{1} {/eq}

Flipping the fraction upside down (making the denominator the numerator and the numerator the denominator) gives us the reciprocal function:

- {eq}\frac{1}{x+5} {/eq}

### How to Find a Reciprocal Function

Finding a reciprocal function is often simply a matter of flipping the fraction upside down. However, the difficulty lies in getting the function into the standard reciprocal function form. The standard reciprocal function form is:

{eq}\frac{a}{x-h} + k {/eq}

Where:

- a is a constant number
- x is the x-variable in the f(x)
- h is the vertical asymptote
- k is the horizontal asymptote

Take a look at the function:

- {eq}f(x)=x+7 {/eq}

The inverse of this function is:

- {eq}\frac{1}{x+7} {/eq}

This is almost in the standard form for reciprocal functions:

- a = 1
- x = x
- h = -7
- k = ?

Since there are no other terms in this equation, it is implied that "k" is 0:

- {eq}\frac{1}{x+7} + 0 {/eq}

### Finding the Standard Reciprocal Form

Some functions are a little harder to find in the standard reciprocal form. For example, the function:

- {eq}\frac{x+5}{4x+21} {/eq}

The inverse function is:

{eq}\frac{4x+21}{x+5} {/eq}

This function does not fit the standard reciprocal function form. There are a couple of methods that can be used to find the standard reciprocal function form:

- Create a table of x and y values for the function and identify the vertical and horizontal asymptotes
- Graph the function (using a graphing tool or by hand) and identify the vertical and horizontal asymptotes

First, create a table of x and y values:

x value | y value |
---|---|

-15 | 3.9 |

-10 | 3.8 |

-5 | Not real number |

0 | 4.2 |

5 | 4.1 |

The vertical asymptote is the x-value when the denominator equals zero; this is illustrated when the y-value is a non-real number (because it doesn't exist). Thus, in this equation, the vertical asymptote is -5. The horizontal asymptote is the number that is being approached in both the positive and negative directions. In this case, 4 is being approached, so the horizontal asymptote is 4.

This means that the reciprocal function in standard form has the following numbers:

- a = 1
- x = x
- h = -5
- k = 4

Making the formula:

- {eq}\frac{1}{x+5} + 4 {/eq}

The other method is looking at a graph of the function:

This graph shows the vertical asymptote at 4 and the horizontal asymptote at -5, giving the same equation as above. The next section discusses what happens if 'a' is not equal to 1 and x is not equal to x.

## Reciprocal Function Graph

The simplest reciprocal function is {eq}\frac{1}{x} {/eq}

In this case:

- a=1
- x=x
- h=0
- k=0

The graph of this function is:

All reciprocal functions have a graph with two sections (called branches), where both curves go off towards infinity towards the vertical and horizontal asymptotes. In this simple reciprocal function, both the horizontal and vertical asymptotes are 0, so the curves are centered around the x and y axis. Increasing the horizontal asymptote will move the entire graph up. Decreasing the horizontal asymptote will move the entire graph down. Increasing the vertical asymptote will move the entire graph to the right while decreasing the vertical asymptote will move the entire graph to the left.

Increasing the value of 'a' will cause the curve to get less steep:

Negative numbers cause the graph to flip:

Numbers between negative one and one will cause the curve to get steeper:

Any changes to the value of x (such as making it 2x) changes the value of 'a'. 2x in the formula really makes 'a' equal to {eq}\frac{1}{2} {/eq} (thus causing the curve to get steeper.

## Reciprocal Function Examples

Let's look at a few reciprocal functions:

- {eq}\frac{2}{x-5}+3 {/eq}

In this reciprocal function:

- a = 2
- h = 5
- k = 3

This means that the curve is half as steep, has moved to the right 5 points, and moved up 3 points:

Now let's look at the reciprocal function:

- {eq}\frac{2}{3x+6}+4 {/eq}

In this reciprocal function:

- a = {eq}\frac{2}{3} {/eq} (the 2 is in the numerator and the 3 is in the denominator with the x)

- h = -2 (when 3 is taken out from the 'x' it also needs to be divided from the 6)
- k = 4

This means that the curve is {eq}\frac{2}{3} {/eq} times steeper, moved to the left 2 points, and moved up 4 points.

### Finding the Domain and Range of the Function

The domain is equal to all numbers that x can be equal to, and the range is equal to all numbers that y can be equal to. This means that the domain is equal to all real numbers except for the vertical asymptote. The range is equal to all real numbers except for the horizontal asymptote.

With the reciprocal function:

{eq}\frac{1}{x-3}+7 {/eq}

The domain is all real numbers except for 3 and the range is all real numbers except for 7:

- Domain: {eq}(-\infty, 3)(3,\infty) {/eq}

- Range: {eq}(-\infty, 7)(7,\infty) {/eq}

## Lesson Summary

The **reciprocal function** is the inverse function or {eq}\frac{1}{f(x)} {/eq} . The general or standard form for the reciprocal function is:

{eq}\frac{a}{x-h} + k {/eq}

In this form:

- a is a constant number, which changes how steep the curve of the graph is
- h is equal to the vertical asymptote
- k is equal to the horizontal asymptote

When a function is not in the standard form, it can be converted by:

- Graphing the function and identifying the vertical and horizontal asymptotes, as well as any changes made to a
- Creating a table of different values of x and the corresponding y values and identifying the vertical and horizontal asymptotes

The graph of a reciprocal function has two branches that curve and go into infinity towards the vertical and horizontal asymptotes.

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