Reciprocal Function

A reciprocal function is just a function that has its variable in the denominator. The concept of reciprocal function can be easily understandable if the student is familiar with the concept of inverse variation as reciprocal function is an example of an inverse variable. Reciprocal functions have a standard form in which they are written. Reciprocals are more than just adding and subtracting. They go beyond that, to division, which can be defined on a graph.

In this article, we are dealing with reciprocal graphs, which are 1s where y is equal to something / x, and here we're representing that something with the letter a. So the a could be any value that you can think of. The most common 1 you'll see though, is y = 1 / x. Let’s see how it is constructed.

To draw it you need to draw a curve in the top right, and then a similar curve in the bottom left. So there are actually 2 separate parts to it even though it is just 1 graph. As well as being able to recognize the graph, you also need to know that it is symmetrical in the slant, angular line that runs across the graph, of y = x because these parts are symmetrical to each others’ parts. And it is also symmetrical in the slant line that runs across the graph at another angle, of y = - x because these parts are symmetrical to each others’ parts.

Pick the x values - 2, 0 and 2. But you could pick any values that appear on your graph. And then we can plug each of these x values into the equation, to find out what the corresponding y values should be.

For example, to find out what y is when x is -2, we just plug -2 into our y = 1 / x equation. So it becomes y = 1 / -2, or just y = minus a half. So we know that when x = - 2 on our graph y should equal - a half which it does.

And finally, if we did the same thing for when x = positive 2, we find that y = positive a half. So because the curve that we were given fits with what we expect from our table of values, we can be fairly sure that it is the y = 1 / x curve.

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.

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 a 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.

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:

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 from 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.

In the above reciprocal graph, we can observe that the graph extends horizontally from -5 to the right side 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 Example of Reciprocal Function - Simplified

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

Solution: The reciprocal of \[y^2 + 6\] is \[\frac{1}{y^2 + 6} \].

The reciprocal of 3y is \[\frac{1}{3y}\].

2. Determine the domain and range of reciprocal function \[y = \frac{1}{x + 6}\] .

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 functions, we will define the inverse of the function by interchanging the position of x and y.

We get,

\[x = \frac{1}{y + 6}\]

Solving the equation for y , we get,

x(y + 6) = 1

xy + 6x = 1

xy = 1 - 6x

\[y = \frac{(1 - 6x)}{x}\]

Therefore, the inverse function is \[y = \frac{(1 - 6x)}{x}\].

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 \[y = \frac{(1 - 6x)}{x}\] 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 \[f(x) = \frac{2}{x - 6}\].

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.

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

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

5. Leonard eats 1/4 of a pizza and divides the remaining into two equal parts for his two sisters. What part of the pizza will each sister receive?

Solution: Part of the pizza eaten by Leonard  = 1/4

Remaining pizza  = 3/4

Given: Remaining pizza is divided into equal parts for his two sisters. 

So, part of the pizza received by each sister is 

=\[\frac{3}{4} \div 2\]

=  \[\frac{3}{(4 \times 2)}\]

=  \[\frac{3}{8}\]

Hence, each sister will receive 3/8 part of the pizza.