What Is Reciprocal Functions - Explanation, Examples & Graphs? Definition, Examples & Key Concepts

A reciprocal function is formed by dividing 1 by a given function. This creates a unique relationship where the original function's output becomes the reciprocal function's input, and vice versa.

A reciprocal function is a function that is the inverse of another.

For example, the reciprocal of 2 is ½.

Reciprocal functions are characterized by their hyperbolic graphs and distinct properties that differentiate them from other function types.

What Is a Reciprocal Function?

Aspect	Details
Definition	A reciprocal function expresses the reciprocal (multiplicative inverse) of a variable.
Components	Consists of a constant in the numerator and an algebraic expression in the denominator.
Simplest Form	f(x)=1/x​
General Form	f(x)= (a​ /x−h) + k
Parameter Descriptions
a	Controls the vertical stretch or compression. A negative a flips the graph across the x-axis.
h	Represents a horizontal shift; the vertical asymptote shifts to x=h.
k	Represents a vertical shift; the horizontal asymptote shifts to y=k.
Transformations	The general form allows for various transformations of the basic reciprocal function f(x)= 1/x

For example: Let's take the function f(x) = 2x + 3. To find its reciprocal function, we follow these steps:

1. Replace f(x) with y: y = 2x + 3
   
2. Swap x and y: x = 2y + 3
   
3. Solve for y: x - 3 = 2y y = (x - 3) / 2
   
4. Replace y with g(x) to represent the reciprocal function: g(x) = (x - 3) / 2

Therefore, the reciprocal function of f(x) = 2x + 3 is g(x) = (x - 3) / 2.

How to Find Reciprocal Functions

Finding the reciprocal of a function is a straightforward process that involves a few key steps:

1. Start with the Original Function: Begin with the function for which you want to determine the reciprocal. Let's call this function f(x).
   
2. Replace f(x) with y: This is simply a notational change to make the subsequent steps easier to manipulate. So, we rewrite f(x) = ... as y = ....
   
3. Interchange x and y: This is the crucial step where we swap the roles of x and y. This reflects the inverse relationship inherent in reciprocal functions. The equation now becomes x = ... (where '...' represents the expression involving y).
   
4. Solve for y: Rearrange the equation to isolate y on one side. This may involve algebraic manipulation depending on the complexity of the original function.

Express the Reciprocal Function: Once you have isolated y, replace it with g(x) to denote the reciprocal function. The final equation, g(x) = ..., represents the reciprocal of the original function f(x).

Example:

Let's illustrate this process with the function f(x) = 3x - 5:

1. Start with f(x): f(x) = 3x - 5
   
2. Replace with y: y = 3x - 5
   
3. Swap x and y: x = 3y - 5
   
4. Solve for y: x + 5 = 3y y = (x + 5) / 3
   
5. Express the reciprocal: g(x) = (x + 5) / 3

Therefore, the reciprocal function of f(x) = 3x - 5 is g(x) = (x + 5) / 3.

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Properties of Reciprocal Functions

Reciprocal functions possess unique properties that set them apart from other types of functions. These properties are crucial for understanding their behavior, sketching their graphs, and more.

Property	Description	Example
Vertical Asymptote	A vertical line (x = a) that the graph approaches but never touches. Occurs where the denominator of the function is zero.	f(x) = 1/(x - 2) has a vertical asymptote at x = 2.
Horizontal Asymptote	A horizontal line (y = b) that the graph approaches as x approaches positive or negative infinity.	f(x) = 2/(x + 1) has a horizontal asymptote at y = 0.
Symmetry	The basic reciprocal function (f(x) = 1/x) is symmetric about the origin.	Rotating the graph of f(x) = 1/x by 180 degrees about the origin results in the same graph.
Domain	All real numbers except for the value(s) that make the denominator zero.	f(x) = 5/(x - 3) has a domain of all real numbers except x = 3.
Range	All real numbers except for the value of the horizontal asymptote.	f(x) = 5/(x - 3) has a range of all real numbers except y = 0.
Non-linearity	Reciprocal functions are curves, not straight lines.	The graph of f(x) = 1/x is a hyperbola.
Discontinuity	Reciprocal functions have a point of discontinuity at the vertical asymptote where the function is undefined.	f(x) = 1/(x - 2) is discontinuous at x = 2.
Inverse Relationship	The reciprocal function and its original function have an inverse relationship; the output of one is the input of the other, and vice versa.	If f(x) = 4x + 1, its reciprocal is g(x) = 1/(4x + 1). f(2) = 9 and g(9) = 1/37.

Graphing of Reciprocal Functions

Reciprocal functions come in various forms, but a fundamental type is represented as k/x, where k is any real number and x is any non-zero value. Let's explore the graphical representation of the simplest case:

f(x) = 1/x 

We'll take a look into plotting this function by selecting different values for x and calculating the corresponding y values.

Understanding the Graph's Characteristics

It's crucial to understand the key features that shape the graph of a reciprocal function:

1. Asymptotes: These are lines that the graph approaches but never intersects.
   • Vertical Asymptote: The line x = 0 (the y-axis) serves as the vertical asymptote because the function is undefined at x = 0 (division by zero is not allowed).
   • Horizontal Asymptote: The line y = 0 (the x-axis) acts as the horizontal asymptote since the function approaches zero as x becomes very large or very small.

2. Branches: The graph of a reciprocal function consists of two separate parts, known as branches, which extend indefinitely towards the asymptotes.

Steps to Graphing f(x) = 1/x

1. Establish the Asymptotes:
   • Draw a dashed vertical line through x = 0 to represent the vertical asymptote.
   • Draw a dashed horizontal line through y = 0 to depict the horizontal asymptote.

2. Calculate and Plot Points:
   • Select a range of whole number x-values, including both positive and negative values: -3, -2, -1, 1, 2, 3.
   • For each chosen x-value, determine the corresponding y-value using the equation y = 1/x. This yields the following points: (-3, -1/3), (-2, -1/2), (-1, -1), (1, 1), (2, 1/2), (3, 1/3).
   • Plot these points accurately on the graph.

x	y
-3	-1/3
-2	-1/2
-1	-1
-1/2	-2
-1/3	-3
1/3	3
1/2	2
1	1
2	1/2
3	1/3

1. Sketch the Curves:
   • Connect the plotted points within each branch using smooth curves.
   • Ensure that the curves approach the asymptotes but never cross them.

Fig: Graph representing the function f(x) = 1/x

Domain and Range of a Reciprocal Function

The domain and range of a reciprocal function describe the possible values for the input (x) and output f(x) of the function, respectively. These are critical to understanding the behavior and constraints of the reciprocal function.

Function Type	Description	Domain (Set Notation)	Range (Set Notation)
f(x) = 1/x	Basic reciprocal function; denominator cannot be zero; output cannot be zero.	x ∈ (−∞,0) ∪ (0,∞)	f(x) ∈ (−∞,0) ∪ (0,∞)
f(x) = a/(x-h) + k	General reciprocal function; denominator (x-h) cannot be zero; output cannot equal the horizontal asymptote (k).	x ∈ (−∞,h) ∪ (h,∞)	f(x) ∈ (−∞,k) ∪ (k,∞)

Expressing Domain and Range

There are different notations to express the domain and range of a function:

• Set Notation: This notation uses curly braces {} and mathematical symbols to define the set of values included in the domain and range.
  
• Interval Notation: This notation uses parentheses () and square brackets [] to indicate the intervals of values included in the domain and range.

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Solved Examples of Reciprocal Functions

Here are a few examples, each showcasing specific concepts, and explanations on how to solve reciprocal functions, including graphing, domain and range, transformations, and asymptotes.

Example 1: Simplest Reciprocal Function

Problem: Find the value of f(x) for f(x) = 1x when:

• x=2
• x= - 3
• x=½ 

Solution:

For f(x) = 1x:

• f(2)= 12
• f(-3) = 1- 3 = - 13
• f(½) = 112 = 2

Example 2: Domain and Range Analysis

Problem: Determine the domain and range of the function f(x) = 1x-4 

Solution:

1. The function is undefined at x−4 = 0 ⇒ x=4
   • Domain: All real numbers except x=4: 

Domain: x ∈ (−∞,4) ∪ (4,∞)

2.  As x approaches infinity or negative infinity, f(x) approaches 0, but never equals it.

• Range: All real numbers except f(x)=0: 

Range: f(x) ∈ (−∞,0) ∪ (0,∞) 

Example 3: Transformation and Asymptotes

Problem: Identify the vertical and horizontal asymptotes of the function     f(x) = 3x+2-1, and sketch the graph.

Solution:

1. Vertical Asymptote: The denominator x+2=0⇒x=−2.
   • Vertical asymptote at x=−2.

2. Horizontal Asymptote: The function approaches y=−1 as x→±∞.
   • Horizontal asymptote at y=−1.

3. Graphing:
   • The graph has two branches, one in the first quadrant and the other in the third quadrant, shifted left by 2 units and down by 1 unit due to the transformations.

Fig: Graph representing the function f(x) = 3x+2-1

Example 4: Value of the Function at Specific Points

Problem: Evaluate f(x) = -2x-1+3 at x=2, x= - 1 and x=1.5

Solution:

For f(x) = -2x-1+3:

1. f(2)= -22-1+3 = -21+3 = -2+3 = 1
2. f(-1)= -2-1-1+3 = -2-2+3 = 1+3 = 4
3. f(1.5)= -21.5-1+3 = -20.5+3 = -4+3 = -1

Example 5: Solving Reciprocal Function Equations

Problem: Solve the equation f(x)=5 for the function f(x) = 4x-2+1

Solution:

1. Set the function equal to 5: 4x-2+1 = 5
2. Subtract 1 from both sides: 4x-2 = 4
3. Multiply both sides by x-2: 4 = 4 (x - 2)
4. Simplify: 4= 4x - 8
5. Add 8 to both sides: 12 = 4x
6. Divide by 4: 3 = x

Thus, the solution is: x=3