# Reciprocal Function

A simple definition of reciprocal is 1 divided by a given number. When we multiply the reciprocal of a number with the number, the result is always 1. Due to this reason, it is also called the **multiplicative inverse.**

The mini-lesson discusses the reciprocal function definition, its domain and range, graphing of the reciprocal function, solved examples on reciprocal functions, and interactive questions.

Try out the reciprocal function calculator given below.

**Lesson Plan**

**What Do You Mean by Reciprocal Functions?**

For a given function \(\begin{align} f(x)\end{align}\), the reciprocal is defined as \(\begin{align} \dfrac{a}{x-h} + k \end{align}\), where the vertical asymptote is \(\begin{align} x=h \end{align}\) and horizontal asymptote is \(\begin{align} y = k \end{align}\). The reciprocal function is also called the "Multiplicative inverse of the function".

The common form of a reciprocal function is \(\begin{align}y = \dfrac{k}{x} \end{align}\), where \(\begin{align}k \end{align}\) is any real number and \(\begin{align}x \end{align}\) can be a variable, number or a polynomial.

The reciprocal of a number is a number which when multiplied with the actual number produces a result of 1

For example, let us take the number \(\begin{align} 2 \end{align}\). The reciprocal is,

When \(\begin{align} \dfrac{1}{2}\end{align}\)

Also, when we multiply the reciprocal with the original number we get 1

\(\begin{align} \dfrac{1}{2} \times 2 = 1\end{align}\)

Some examples of reciprocal functions are,

\(\begin{align}f(x) &= \dfrac{1}{5} \\ f(x) &= \dfrac{2}{x^2} \\ f(x) &= \dfrac{3}{x-5}\end{align}\)

In the **exponent form**, the **reciprocal function** is written as,

\(\begin{align}f(x) = a(x-h)^{-1} + k \end{align}\) |

**Characteristics of Reciprocal Functions**

- Reciprocal functions are in the form of a fraction. The numerator is a real number and the denominator is either a number or a variable or a polynomial.
- The reciprocal \(\begin{align} x \end{align}\) is \(\begin{align} \dfrac{1}{x}\end{align}\)
- The denominator of a reciprocal function cannot be 0. For example, \(\begin{align} f(x) = \dfrac{3}{x-5}\end{align}\) cannot be 0, which means 'x' cannot take the value 5.
- The domain and range of the reciprocal function \(\begin{align} f(x) = \dfrac{1}{x}\end{align}\) is the set of all real numbers except 0.
- The graph of the equation \(\begin{align} f(x) = \dfrac{1}{x}\end{align}\) is symmetric with the equation \(\begin{align} y = x \end{align}\).

**How to Graph Reciprocal Functions?**

There are many forms of the reciprocal functions. One of them is of the form \(\begin{align} \dfrac{k}{x}\end{align}\). Here 'k' is real number and the value of 'x' cannot be 0. Now let us draw the graph for the function \(\begin{align}f(x) = \dfrac{1}{x} \end{align}\) by taking different values of x and y.

x | -3 | -2 | -1 | -1/2 | -1/3 | 1/3 | 1/2 | 1 | 2 | 3 |

y | -1/3 | -1/2 | -1 | -2 | -3 | 3 | 2 | 1 | 1/2 | 1/3 |

For a reciprocal function \(\begin{align} f(x) = \dfrac{1}{x} \end{align}\), 'x' can never be 0 and so \(\begin{align} \dfrac{1}{x} \end{align}\) can also not be equal to 0. Therefore the domain and range of reciprocal function are as follows.

\(\begin{align}{\{x \in R\: | \:x \neq 0\}} \end{align}\)
\(\begin{align}{\{x \in R\: | \:x \neq 0\}} \end{align}\) |

From the graph we observe that they never touch the x-axis and y-axis.

The y-axis is said to be the vertical asymptote as the curve gets very closer but never touches it.

Also, the x-axis is the horizontal asymptote as the curve never touches the x-axis.

- Sketch the graph of the reciprocal function f(x) = -1/x and find how it is related to the function f(x) = 1/x.

**How Do You Solve Reciprocal Functions?**

**Reciprocal of a Number **

To find the reciprocal we divide the number, variable, or expression by 1

For example,

Reciprocal of 6 is \(\begin{align} &\dfrac{1}{6}\end{align}\)

**Reciprocal of a Variable**

The reciprocal of a variable 'y' can be found by dividing the variable by 1

For example,

Reciprocal of y is \(\begin{align} &\dfrac{1}{y}\end{align}\)

**Reciprocal of a Expression**

The reciprocal of an expression can be found by exchanging the positions of numerator and denominator.

Examples are,

Reciprocal of \(\begin{align}\dfrac{x}{x-4}\end{align}\) is \(\begin{align}\dfrac{x-4}{x}\end{align}\)

**Reciprocal of a Fraction**

Reciprocal of a fraction can be obtained by flipping the places of numerator and denominator.

For example,

Reciprocal of \(\begin{align}\dfrac{5}{8}\end{align}\) is \(\begin{align}\dfrac{8}{5}\end{align}\).

**Reciprocal of a Mixed Fraction**

Reciprocal of a mixed fraction can be obtained by finding the improper fraction and then finding its reciprocal.

For example,

To find the reciprocal of \(\begin{align}3\dfrac{3}{4}\end{align}\), we find the improper fraction which is \(\begin{align}\dfrac{15}{4}\end{align}\) and now find the reciprocal \(\begin{align}\dfrac{4}{15}\end{align}\)

- For a function f(x), 1/f(x) is the reciprocal function.
- Reciprocal is also called as the multiplicative inverse.
- The reciprocal function y = 1/x has the domain as the set of all real numbers except 0 and the range is also the set of all real numbers except 0.
- An asymptote is a line that approaches a curve but does not meet it. For the reciprocal function f(x) = 1/x, the horizontal asymptote is the x-axis and vertical asymptote is the y-axis.
- The vertical asymptote is connected to the domain and the horizontal asymptote is connected to the range of the function.

**Solved Examples**

Example 1 |

Find the reciprocal of the following.

a) \(\begin{align}5\end{align}\)

b) \(\begin{align}3x\end{align}\)

c) \(\begin{align}x^2 + 6\end{align}\)

d) \(\begin{align}\dfrac{5}{8}\end{align}\)

**Solution**

Reciprocal of a number or a variable 'a' is 1/a.

and

Reciprocal of a fraction 'a/b' is 'b/a'.

Reciprocal of \(\begin{align}5\end{align}\) is \(\begin{align}\dfrac{1}{5}\end{align}\) Reciprocal of \(\begin{align}3x\end{align}\) is \(\begin{align}\dfrac{1}{3x}\end{align}\) Reciprocal of \(\begin{align}x^2+6\end{align}\) is \(\begin{align}\dfrac{1}{x^2+6}\end{align}\) Reciprocal of \(\begin{align}\dfrac{5}{8}\end{align}\) is \(\begin{align}\dfrac{8}{5}\end{align}\) |

Example 2 |

Find the domain and range of the reciprocal function \(\begin{align}y = \dfrac{1}{x+3}\end{align}\)

**Solution**

To find the domain of the reciprocal function, let us equate the denominator to 0

\(\begin{align}x+3 = 0\end{align}\) \(\begin{align}\therefore x = -3\end{align}\)

So, the domain is the set of all real numbers except the value \(\begin{align}x = -3\end{align}\).

The range of the reciprocal function is the same as the domain of the inverse function.

To find the range of the function let us define the inverse of the function, by interchanging the places of \(\begin{align}x\end{align}\) and \(\begin{align}y\end{align}\).

We get,

\(\begin{align}x = \dfrac{1}{y+3}\end{align}\).

Solving the equation for y, we get,

\(\begin{align}x (y+3)= 1\end{align}\) \(\begin{align}xy+3x = 1\end{align}\) \(\begin{align}xy= 1-3x\end{align}\) \(\begin{align}y = \dfrac {1-3x}{x}\end{align}\)

The inverse function is \(\begin{align}y = \dfrac {1-3x}{x}\end{align}\)

Now equating the denominator to 0 we get,

\(\begin{align}x= 0\end{align}\).

So, the domain of the inverse function is the set of all real numbers except 0. Since the range is also the same, we can say that,

the range of the function \(\begin{align}y = \dfrac{1}{x+3}\end{align}\) is the set of all real numbers except 0

Domain is set of all real numbers except the value \(\begin{align}x = -3\end{align}\). Range is the set of all real numbers except \(\begin{align}0\end{align}\). |

Example 3 |

Find the vertical and horizontal asymptote of the function \(\begin{align}f(x) = \dfrac {2}{x-7}\end{align}\)

**Solution**

To find the vertical asymptote take the denominator and equate it to 0

We get,

\(\begin{align}x-7 = 0\end{align}\)

Therfore the vertical asymptote is \(\begin{align} x = 7\end{align}\).

To find the horizontal asymptote we need to consider the degree of the polynomial of the numerator and the denominator. Since the numerator's degree is less than the denominator the horizontal asymptote is 0.

The vertical asymptote is \(\begin{align} x = 7\end{align}\). The horizontal asymptote is \(\begin{align} y= 0\end{align}\). |

**Interactive Questions**

**Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.**

**Let's Summarize**

The mini-lesson targeted the fascinating concept of reciprocal functions. The math journey around reciprocal function starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that is not only relatable and easy to grasp but will also stay with them forever. Here lies the magic with Cuemath.

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**Frequently Asked Questions (FAQs)**

### 1. What is the equation of reciprocal equation?

f(x) = 1/x is the equation of reciprocal equation.

### 2. What is the inverse of reciprocal equation?

\(\begin{align} f^{-1}(x)\end{align}\) is the inverse of the reciprocal equation \(\begin{align} f(x)\end{align}\).

### 3. Is a reciprocal function continuous?

Yes, the reciprocal function is continuous at every point other than the point at x =0

### 4. Is a reciprocal function bounded?

Since the reciprocal function is uniformly continuous, it is bounded.

### 5. What is the end behavior of a reciprocal function?

The end behavior of a reciprocal function describes the value of 'x' in the graph approaching negative infinity on one side and positive infinity on the other side.

### 6. Is the reciprocal function a polynomial?

Yes, a polynomial is a self-reciprocal.

### 7. How do you find the reciprocal of a quadratic function?

By factoring and finding the x-intercepts of a quadratic equation(It may be zero, one, or two) we can find the reciprocal of a quadratic equation.

### 8. How do you find the invariant part of a quadratic function?

The points f(x) = 1 and f(x) = -1 are called the invariant points of the reciprocal function. The reciprocal function graph always passes through these points.

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