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# Reciprocal Function – Properties, Graph, and Examples

Finding the reciprocal function will return a new function â€“ the reciprocal function. These functions exhibit interesting properties and unique graphs. When two expressions are inversely proportional, we also model these behaviors using reciprocal functions.**From its definition to its graph, weâ€™ll learn extensively about reciprocal functions in this article. Weâ€™ll learn the following:**

*Reciprocal functions consist of two components: a constant on the numerator and an algebraic expression in the denominator.*- Understanding the properties of reciprocal functions.
- Graphing reciprocal functions using different methods.
- Determining the functionâ€™s expression based on its graph.

## What is a reciprocal function?

The common form of reciprocal functions that we may encounter is $y = \dfrac{k}{x}$, where $k$ is a real number. This means that reciprocal functions are functions that contain constant on the numerator and algebraic expression in the denominator. Here are some examples of reciprocal functions:- $f(x) = \dfrac{2}{x^2}$
- $g(x) = \dfrac{1}{x + 1} â€“ 4$
- $h(x) = -\dfrac{2}{x + 4} + 3$

## How to find the reciprocal of a function?

As we have learned in the past, we can determine a number’s reciprocal by dividing 1 by the given number. The same concept applies when we find a function’s reciprocal function â€“**we divide 1 by the functionâ€™s expression**. Hereâ€™s a table to compare the reciprocal that we learned in the past and reciprocal functions:

Reciprocal |
Reciprocal Function |

Given a number, $k$, its reciprocal is $\dfrac{1}{k}$. | Given a function, $f(x)$, its reciprocal function is $\dfrac{1}{f(x)}$. |

The product of $k$ and its reciprocal is equal to $k$ Â· $\dfrac{1}{k} = 1$. | The product of $f(x)$ and its reciprocal is equal to $f(x)$ Â· $\dfrac{1}{f(x)} = 1$. |

Given $\dfrac{1}{k}$, its value is undefined when $k = 0$. | Given $\dfrac{1}{f(x)}$, its value is undefined when $f(x) = 0$. |

- This means that if we want to find the reciprocal of $y = 2x â€“ 1$, its reciprocal can be expressed as $y = \dfrac{1}{2x â€“ 1}$.
- We can also confirm the product of $2x â€“ 1$ and its reciprocal:

- This also means that $2x â€“ 1$ must never be zero, so $x$ must never be $\frac{1}{2}$.

## How to graph reciprocal functions?

There are different ways for us to graph reciprocal functions. In this article, weâ€™ll focus on two methods:- Graphing reciprocal functions by finding the functionâ€™s table of values first.
- Graphing reciprocal functions using different transformation techniques.

### How to graph functions using their tables of values

Letâ€™s go ahead and start with graphing the parent function, $y = \frac{1}{x}$ by first finding its table of values.- Find the value of the function at different values of $x$.
- Plot these points on the $xy$-coordinate system.
- Graph the curves using these points.

$\mathbf{x}$ | Â $-3$ | $-2$ | $-1$ | $-\frac{1}{2}$ | $-\frac{1}{3}$ | $\frac{1}{3}$ | $\frac{1}{2}$ | $1$ | $2$ | $3$ |

$\mathbf{y}$ | $-\frac{1}{3}$ | $-\frac{1}{2}$ | $-1$ | $-2$ | $-3$ | $3$ | $2$ | $1$ | $\frac{1}{2}$ | $\frac{1}{3}$ |

Domain |
$(-\infty, 0) \cup (0, \infty)$ |

Range |
$(-\infty, 0) \cup (0, \infty)$ |

### How to graph reciprocal functions by transformation

Here are some common graph transformations that weâ€™ve already learned in the past:- Vertically and horizontally translating functions.
- Vertically and horizontally stretching functions.
- Shrinking functions vertically and horizontally.

- Translate the graph one unit to the right.
- Stretch the graph vertically by two units.
- Translate the graph four units upward.

Vertical Asymptote |
Horizontal Asymptote |

The expression $\frac{2}{x-1}$ canâ€™t be $0$, so $y \neq 4$. Asymptote:$ y = 4$ | The denominator $x â€“ 1$ canâ€™t be $0$, so $x \neq 1$. Asymptote: $x = 1$ |

**vertical asymptote can be expressed as y = k,**and the

**horizontal asymptote can be expressed as x = h**. From the graph and the asymptotes, we can also find the functionâ€™s domain and range:

Domain |
$(-\infty, 1) \cup (1, \infty)$ |

Range |
$(-\infty, 4) \cup (4, \infty)$ |

- Try graphing $y = -\dfrac{1}{x}$ on your own and compare this with the graph of $y = \dfrac{1}{x}$.
- Sketch $y = x$ and $y = -x$ on the graphs of $y = \dfrac{1}{x}$ and $-\dfrac{1}{x}$.
- What can you say about each pair of graphs?

**the function is positive, it is symmetric with respect to the equation**$\mathbf{y = x}$. Meanwhile, when the

**function is negative (i.e., has a negative constant), it is symmetric with respect to the equation**$\mathbf{y = -x}$.

### Summary of reciprocal function definition and properties

Before we try out some more problems that involve reciprocal functions, letâ€™s summarize everything that we have learned so far about these unique functions.- Reciprocal functions are functions that have a constant on their denominator and a polynomial on their denominator.
- The reciprocal of a function, $f(x)$, can be determined by finding the expression for $\dfrac{1}{f(x)}$.
- We can graph a reciprocal function using the functionâ€™s table of values and transforming the graph of $y = \dfrac{1}{x}$.
- Make sure to find the vertical and horizontal asymptotes of the function.
- The domain and range of a reciprocal function will depend on the asymptotesâ€™ values.
- The symmetry of the reciprocal functionâ€™s graph will depend on the constantâ€™s sign.

**If $g(x)$ is the reciprocal of $f(x)$, what is the value of $g(x) \cdot f(x)$?**

*Example 1*__Solution__Find the expression for $g(x)$ in terms of $f(x)$. Since it is the reciprocal of $f(x)$, we have $g(x) = \frac{1}{f(x)}$. Multiply the two expressions to find their product. $\begin{aligned} g(x) \cdot f(x)&= \frac{1}{f(x)}\cdot f(x)\\&=\mathbf{1}\end{aligned}$ We can also use the fact that

**a function’s productÂ and its reciprocal will always be equal to 1**.

**What are the reciprocals of the following functions? a. $f(x) = Â x^2 â€“ 4$ b. $g(x) = \dfrac{x}{2}$ c. $h(x) = \dfrac{3(x – 1)}{5}$**

*Example 2*__Solution__To find the reciprocal of a function, we can

**divide 1 by the expression**. a. For $f(x) = x^2 â€“ 4$, we simply divide $1$ by the $f(x)$â€™s expression, hence its reciprocal is equivalent to $\mathbf{\dfrac{1}{x^2-4}}$. b. When finding the reciprocal of a rational expression, we

**switch the numerator and denominator positions**. This means that its reciprocal is $\mathbf{\dfrac{2}{x}}$. We apply the same process when finding the reciprocal of $h(x)$. Hence, $\frac{1}{h(x)}\mathbf{=\dfrac{5}{3(x-1)}}$.

**Complete the table below by finding the reciprocal functionsâ€™ symmetry, domain, range, vertical asymptote, and horizontal asymptote.**

*Example 3*Function |
Symmetry |
Vertical Asymptote |
Horizontal Asymptote |
Domain |
Range |

$y=-\dfrac{3}{x}$ | |||||

$y=\dfrac{2}{2x+1}$ | |||||

$y=\dfrac{4}{x-4}+2$ | |||||

$y=-\dfrac{3}{x-2}+5$ |

__Solution__When finding the symmetry of the reciprocal function, we base it on the constantâ€™s sign.

- If the constant is negative, its graph is symmetric with respect to the line $y = -x$.
- If the constant is positive, the graph is symmetric with respect to $y = x$.

- The vertical asymptote will be $x = h$.
- The horizontal asymptote will be $y = k$.

Function |
Symmetry |
Vertical Asymptote |
Horizontal Asymptote |
Domain |
Range |

$y=-\dfrac{3}{x}$ | $y = -x$ | $x = 0$ | $y = 0$ | $(-\infty,0)\cup(0, \infty)$ | $(-\infty,0)\cup(0, \infty)$ |

$y=\dfrac{2}{2x+1}$ | $y = x$ | $x = 0$ | $y = 1$ | $(-\infty,0)\cup(0, \infty)$ | $(-\infty,0)\cup(0, \infty)$ |

$y=\dfrac{4}{x-4}+2$ | $y = x$ | $x = 4$ | $y = 2$ | $(-\infty,4)\cup(4, \infty)$ | $(-\infty,2)\cup(2, \infty)$ |

$y=-\dfrac{3}{x-2}+5$ | $y = -x$ | $x = 2$ | $y = 5$ | $(-\infty,2)\cup(2, \infty)$ | $(-\infty,5)\cup(5, \infty)$ |

**Choose from the four functions given in Example 3 and graph the two of them:**

*Example 4*- One function is to be graphed by finding the table of values.
- The second function is to be graphed by transforming $y=\dfrac{1}{x}$.

$\mathbf{x}$ | $\mathbf{y = -\dfrac{3}{x}}$ |

$\dfrac{1}{3}$ | $-9$ |

$\dfrac{1}{2}$ | $-6$ |

$1$ | $-3$ |

$3$ | $-1$ |

- Translate $y = \dfrac{1}{x}$ to the right by $4$ units.
- Vertically stretch the functionâ€™s graph by $4$.
- Translate the resulting function by $2$ units upwards.

Domain |
$(-\infty,4)\cup(4, \infty)$ |

Range |
$(-\infty,2)\cup(2, \infty)$ |

### Practice Questions

*Images/mathematical drawings are created with GeoGebra.*

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