Reciprocal Function

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What are Reciprocal Functions?

The most common form of reciprocal function that we observe is y = k/z, where the variable k is any real number. It implies that reciprocal functions are functions that have constant in the numerator and algebraic expression in the denominator. Here are some examples of reciprocal functions:

\[f(x) = \frac{5}{x^{2}}\]

\[g(x) = \frac{2}{x + 1} - 4\]

\[h(x) = \frac{-3}{x + 4} + 2\]

As we can see in all the reciprocal functions examples given above, the functions have numerators that are constant and denominators that include polynomials.

The general form of reciprocal function equations is \[f(x) = \frac{a}{x - h} + k\], where a, h, and k are real numbers constant.


Reciprocal Function Equation

The general form of reciprocal function equation is given as \[f(x) = \frac{a}{x - h} + k\].

Where the variables a,h, and k are real numbers constant.


How to do Reciprocal Function?

The reciprocal of a number can be determined by dividing the variable by 1. Similarly, the reciprocal of function is determined by dividing 1 by the function's expression.


Example:

  • Given a function f(y) , its reciprocal function is 1/f(y).

  • The product of f(y), and its reciprocal function is equal to  f(y).1/f(y) = 1.

  • Given, 1/f(y), its value is undefined when f(y)= 0.

How to Construct a Reciprocal Function Graph

There are different forms of reciprocal functions. One of the forms is k/x, where k is a real number and the value of the denominator i.e. x cannot be 0.

Now, let us draw the reciprocal graph for the function f(x) = 1/x by considering the different values of x and y.


Value of x

-3

- 2

-1

-1/2

-1/3

1/3

1/2

1

2

3

Value of y

-1/3

-1/2

-1

-2

-3

3

2

1

1/2

1/3


reciprocal graph for function f(x) = 1/x


For a given reciprocal function f(x) = 1/x, the denominator ‘x’ cannot be zero, and similarly, 1/x can also not be equal to 0.


Therefore, the reciprocal function domain and range are as follows:


The domain is the set of all real numbers excluding 0, as 1/x is undefined.

Domain = { x R| x ≠ 0

Similar to the domain, the range is also the set of all real numbers. Therefore,

Range = { x R| x ≠ 0


From the reciprocal function graph, we can observe that the curve never touches the x-axis and y-axis. 

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

Similarly, the x-axis is considered to be a horizontal asymptote as the curve never touches the x-axis.


How to Find Range and Domain of Reciprocal Function Form a Graph?

The domain is the set of all possible input values. The domain of a graph includes all the input values shown on the x-axis whereas the range is the set of all possible output values. If the reciprocal function graph continues beyond the portion of the graph, we can observe the domain and range may be greater than the visible values.


reciprocal graph


In the above reciprocal graph, we can observe that the graph extends horizontally from -5 to the right side without beyond.

Note: The reciprocal function domain and range are also written from smaller to larger values, or from left to right for the domain, and from the bottom of the graph to the of the graph for range.


Facts to Remember

  • The reciprocal of x = 1/x.

  • The reciprocal is also known as the multiplicative inverse

  • The domain and range of the reciprocal function x = 1/y is the set of all real numbers except 0.

  • An asymptote in a reciprocal function graph is a line that approaches a curve but does not touch it. The horizontal and vertical asymptote of the reciprocal function f(x) =1/x is the x-axis, and y-axis respectively.

  • The vertical asymptote of the reciprocal function graph is linked to the domain whereas the horizontal asymptote is linked to the range of the function.

Solved Examples

1. Find the reciprocal of y² + 6 and 3y 

Solution:

  • The reciprocal of y² + 6 is 1/y² + 6 .

  • The reciprocal of 3y is 1/3y.

2. Determine the domain and range of reciprocal function y = 1/x + 6 .

Solution:

To find the domain and range of reciprocal function, the first step is to equate the denominator value to 0. Accordingly,

x + 6 = 0

x = - 6

So, the domain of the reciprocal function is the set of all real numbers except the value x = -6.

The range of the reciprocal function is similar to the domain of the inverse function.

To find the range of reciprocal function, we will define the inverse of the function by interchanging the position of x and y.

We get.

x = 1/y + 6

Solving the equation for y , we get,

x(y + 6) = 1

xy + 6x = 1

xy = 1 - 6x

y = (1 - 6x)/x

Therefore, the inverse function is y = (1 - 6x)/x.

Now, equating the denominator value, we get x = 0.

Thus, the domain of the inverse function is defined as the set of all real numbers excluding 0. As the range is similar to the domain, we can say that,

The range of the function y = 1/x + 6 is the set of all real numbers except 0.

Accordingly,

The domain is the set of all real numbers except the value x = - 6, whereas the range is the set of all real numbers except 0.


3. Find the horizontal and vertical asymptote of the function f(x) = 2/x - 6.

Solution:

To find the vertical asymptote we will first equate the denominator value to 0.

We get, x - 6 = 0

Therefore, the vertical asymptote is x = 6.

To find the horizontal asymptote, we need to observe the degree of the polynomial of both numerator and denominator. As the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is 0.

Accordingly,

The vertical asymptote is x = 6.

The horizontal asymptote is y = 0.


4. Find the domain and range of the function f in the following graph.

Solved Example 4

Solution:  In the above graph, we can observe that the horizontal extent of the graph is  -3 to 1. Hence, the domain f is [-3,1].

Solution for Solved Example 4

The vertical extent of the above graph is 0 to -4. Hence the range is [-4.0].

FAQ (Frequently Asked Questions)

1. What are the Characteristics of Reciprocal Function?

Ans:

  • Reciprocal functions are expressed in the form of a fraction. A numerator is a  real number, whereas the denominator is a number, variable, or expression.

  • The reciprocal of y is 1/y.

  • The denominator of reciprocal function can never be 0. For example, f(y) = 3/(y - 5), which implies that ‘y’ cannot take the value 5.

  • The reciprocal function domain and range f(y) = 1/y is the set of all real numbers except 0.

  • The graph of the equation f(y) = 1/y is symmetric with equation x = y.

2. What are the Characteristics of the Reciprocal Function Graph?

Ans: The graph of reciprocal function y = k/x get closer to the x-axis. as the value of x increases, but it never touches the x-axis. The is known as the horizontal asymptote of the graph.


Each point of the graph gets close to the y = axis as the value of x gets closer to 0 but never touches the y - axis because the value of y cannot be defined when x = 0. This Is known as the vertical asymptote of the graph. This type of curve is known as a rectangular hyperbola.

3. What is the Standard Form of Reciprocal Function Equation?

Ans: The standard form of reciprocal function equation is given as f(x) = a/(x - h) + k.