Math Lesson 15.4.1 - The Meaning of the Term "Reciprocal" in Math

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Welcome to our Math lesson on The Meaning of the Term "Reciprocal" in Math, this is the first lesson of our suite of math lessons covering the topic of Reciprocal Graphs, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

The Meaning of the Term "Reciprocal" in Math

Before dealing with the reciprocal graphs, let's explain what the term "reciprocal" means in mathematics. Let's explain this by illustrating with numbers. Thus, in mathematics, the reciprocal of a given number a is 1/a. For example, the reciprocal of 4 is 1/4, the reciprocal of 12 is 1/12 and so on. From this definition, it is clear that multiplying a number a and its reciprocal 1/a gives always 1.

For example,

and so on.

Example 1

Find the value of x that make the following pairs of numbers reciprocal

1. 3 - 2x and 1/5
2. 1/2x and 6
3. 1/1 - 3x and 1/10

Solution 1

From theory, we already know that the condition for two numbers to be reciprocal is that when multiplied, their product must give -1. Thus, we have

1. (3 - 2x) ∙ 1/5 = 1
   Multiplying both sides by 5 yields
   5 ∙ (3 - 2x) ∙ 1/5 = 5 ∙ 1
   3 - 2x = 5
   -2x = 5 - 3
   -2x = 2
   x = 2/-2
   x = -1
   Indeed,
   3 - 2 ∙ (-1)
   = 3 + 2
   = 5
   5 and 1/5 are reciprocal
2. 1/2x ∙ 6 = 1
   6/2x = 1
   Multiplying both sides by 2 yields
   2 ∙ 6/2x = 2 ∙ 1
   6/x = 2
   x = 6/2
   x = 3
   Indeed,
   1/2 ∙ 3 = 1/6
   1/6 and 6 are reciprocal
3. 1/1 - 3x ∙ 1/10 = 1
   1/(1 - 3x) ∙ 10 = 1
   (1 - 3x) ∙ 10 = 1
   1 ∙ 10 - 3x ∙ 10 = 1
   10 - 30x = 1
   -30x = 1 - 10
   -30x = -9
   x = -9/-30
   x = 9/30
   x = 3/10
   Indeed,
   1/1 - 3 ∙ 3/10
   = 1/1 - 9/10
   = 1/10/10 - 9/10
   = 1/1/10
   = 1 ∙ 10/1
   = 10
   10 and 1/10 are reciprocal

These are the following things to consider when dealing with reciprocals:

1. The reciprocal of a number a is 1/a.
2. The reciprocal of a variable x is 1/x.
3. The reciprocal of an expression E(x) is 1/E(x). For example, the reciprocal of 2x + 1 is 1/2x + 1
4. The reciprocal of 1 is 1 because 1/1 = 1.
5. The number 0 has no reciprocal as 1/0 is undefined. However, in the proximity of 0 we can assume the reciprocal of 0 to be + ∞. This is because when the denominator of a fraction gets closer and closer to 0 the fraction's value increases more and more.
6. The reciprocal of a negative number -n is -1/n.
7. The reciprocal of a fraction a/b is b/a. This is because 1/a/b = 1 × (b/a) = b/a.
8. The reciprocal of a mixed number is found by converting the mixed number into an improper fraction first, and then applying the 7th point. For example, the reciprocal of
   3 2/5
   is found after converting this mixed number into an improper fraction, i.e.
   3 2/5 = 3 × 5 + 2/5 = 15 + 2/5 = 17/5
   Given that the reciprocal of 17/5 is 5/17, we say the reciprocal of
   3 2/5 is 5/17
9. The reciprocal of a decimal n is 1/n. However, this is not sufficient, as it is not appropriate to write the denominator of a fraction as a decimal. Therefore, the next step involves multiplying both the numerator and denominator of the reciprocal obtained by a suitable number that allows getting rid of the decimal point. Then, we make any further simplification if necessary. For example, since the reciprocal of 3.2 is 1/3.2, we write
   1/3.2 = 1 × 10/3.2 × 10 = 10/32 = 5/16

Example 2

Find the reciprocal of the following numbers. Write them in the simplest form.

1. 4/8
2. -2/5
3. 3 - 5x
4. 5.24
5. 4 2/7

Solution 2

1. If we write a = 4/8, then its reciprocal is
   1/a = 1/4/8/
   = 1 × 8/4
   = 8/4
   = 2
2. If we write b = -2/5, then its reciprocal is
   1/b = 1/-2/5
   = 1 × (-5/2)
   = -5/2
3. If we write c = 3 - 5x, then its reciprocal is
   1/c = 1/3 - 5x
4. If we write d = 5.24, then its reciprocal is
   1/d = 1/5.24
   = 1 × 100/5.24 × 100
   = 100/524
   = 25/131
5. If we write e = 4 2/7, first we express this mixed number into an improper fraction, i.e.
   e = 4 2/7
   = 4 × 7 + 2/7
   = 28 + 2/7
   = 30/7
   then, we write the reciprocal of this number as
   1/e = 1/30/7/
   = 1 × 7/30
   = 7/30

Definition of Reciprocal Functions

Now that the meaning of the term "reciprocal" is clear, let's explain what a reciprocal function represents. Thus, the basic (simplest) form of a reciprocal function occurs when the product of the two variables contained in it is equal to 1. In this way, we say the two variables x and y are reciprocal to each other.

In symbols, we write

However, given that in functions the variable y is written on the left side and the rest of the terms on the right side, we transform the above formula as

We have already seen this type of relationship between variables in chapter 4 when explaining various types of proportion. Thus, in that chapter, we have seen that the formula above gives an inverse (indirect) relationship between the variables, which we have called an "inverse proportion". We have seen that in situations involving inverse proportions, when one of the quantities increases by a certain factor, the other quantity decreases by the same factor. The product of the two quantities involved was always constant in such proportions. Therefore, we say a reciprocal function represents the relationship between two variables in inverse proportion to each other.

As an extension of the meaning of reciprocal numbers (where the product must always be equal to 1), we include in the set of reciprocal functions all functions that represent an inverse relationship between the variables, i.e. all those functions in which the product of the variables is always the same number, albeit not 1. For example, since

are all examples of the inverse relationship between the variables, all the corresponding functions derived from such a relationship

are also considered as reciprocal functions.

Moreover, we will extend the set of reciprocal functions to include situations in which the relationship between variables is not exactly like that of an inverse (indirect) proportion, i.e. where any increase in the values of one variable does not bring a decrease in the values of the other variable by exactly the same factor. For example, we will consider

as a reciprocal function despite not having a pure inverse relationship between the variables. Hence, for x₁ = 1 and x² = 3, we have an increase in the x-values by a factor of 3. This increase does not bring a decrease of the y-values by the same factor, i.e. by 3 times, because

and

In a pure inverse relationship, we'd expect the y-value to decrease 3 times, i.e. from 1 it would become 1/3, not 1/4 as occurred here. However, since the graph of this function is similar to that of all reciprocal functions (as we are going to see later on in this tutorial), we consider functions of this kind as reciprocal. Thus, all the following functions

are also reciprocal, as all of them contain the independent variable in the denominator of a fraction and have it expressed to the first power, just like the basic form of the reciprocal function.

Now, look at the third function above. Apparently, it seems that no x-variable is in the numerator. However, doing some transformations to include the constant 3 within the fraction yields

In this way, we find out that a reciprocal function may contain its independent variable not only in the denominator but also in the numerator in certain cases, (when either the denominator or the numerator contains a constant term besides the first-degree monomial). Therefore, despite the basic form of a reciprocal function does not contain variables in the numerator, don't hurry up to exclude a function from the set of reciprocal ones, as perhaps you can remove the variable from the numerator by doing some transformations in the original function.

Example 3

Which of the following function(s) is/are reciprocal?

1. y = 2x - 3/4x - 6
2. y = x/x - 1 + 1
   y = 1/3x + x
3. y = x² - 3x + 2/x(x - 2)

Solution 3

Let's try to transform all functions in such a form that allows us to better understand whether they are reciprocal or not. Thus, we have:

1. y = 2x - 3/4x - 6
   = 2x - 3/2 ∙ (2x - 3)
   We can simplify the like terms for 2x - 3 ≠ 0, i.e. for x ≠ 3/2. In this way, we obtain
   y = 1/2
   which is a constant function (hence, not a reciprocal one).
2. y = x/x - 1 + 1
   = x/x - 1 + x - 1/x - 1
   = x + x + 1/x - 1
   = 2x + 1/x - 1
   Hence, this function is reciprocal for x ≠ 1, as it has the variable x combined with constant terms in at least one of the parts of the fraction.
3. y = 1/3x + x
   = 1/3x + 3x²/3x
   = 3x² + 1/3x
   This function is not reciprocal, as the numerator contains a quadratic expression.
4. y = x² - 3x + 2/x(x - 2)
   = x² - 2x + 1-x + 1/x(x - 2)
   = (x²-2x + 1) - (x - 1)/x(x - 2)
   = (x - 1)² - (x - 1)/x(x - 2)
   = (x - 1)(x - 1 - 1)/x(x - 2)
   = (x - 1)(x - 2)/x(x - 2)
   = x - 1/x
   This function is reciprocal for x ≠ 2, as both terms in the fraction contain polynomials to the first power at maximum.

Now is a good point to summarize everything we have explained so far and to provide a more precise definition for reciprocal functions.

A reciprocal function is a function that after completing all the necessary operations can be written in the form

where a, h and k are numbers and x is the independent variable.

In simpler forms as those we have seen earlier, one or more elements of the above formula may not be present. We think about them as being zero. For example, in the simplest form of a reciprocal function

we have a = 1, h = 0 and k = 0.

In this way, the first thing to do when checking whether a function is reciprocal or not is to try to write it in the standard form

Let's consider another example to clarify this point.

Example 4

Check whether the following functions are reciprocal or not by attempting to express them in the form

In each case when the function is reciprocal, identify the values of a, h and k.

1. y = x + 6/x² + 5x - 6 + 3/x - 1
2. y = x² - 2x + 1/x - 1 + x² - 1/x + 1

Solution 4

1. We have
   y = x + 6/x² + 5x - 6 + 3/x - 1
   = x + 6/(x - 1)(x + 6) + 3/x - 1
   = 1/x - 1 + 3/x - 1
   = 4/x - 1
   = 4/x - 1 + 0
   This is a reciprocal function where a = 4, h = 1 and k = 0.
2. We have
   y = x² - 2x + 1/x - 1 + x² - 1/x + 1
   = (x - 1)²/x - 1 + (x - 1)(x + 1)/x + 1
   = x - 1 + x - 1
   = 2x - 2
   This is not a reciprocal function but a linear one instead.

You have reached the end of Math lesson 15.4.1 The Meaning of the Term "Reciprocal" in Math. There are 5 lessons in this physics tutorial covering Reciprocal Graphs, you can access all the lessons from this tutorial below.

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