Inverse Relation

An inverse relation simply exchanges every pair in a relation; it is obtained by interchanging the elements of every given pair in the given relation. If R it is a relation from set A to set B, then defined as R = {(x, y): x A yA}.
R={(x, y):xA, yB}

If R is a relation from the set A to set B, then the inverse relation is
R-1={(y, x):yB,xA}
 

If the relation R is related to an item from set A to set B, then its reverse relation R-1connects elements from set B back to set A by swapping each pair.

For every (x, y) R, you get (y, x) R-1
In set terms: R AB, R-1  BA

If a relation is shown as a graph, its inverse can be found by reflecting it across the line y = x. Pick the points (x, y) from the graph, swap them to (y,x), and plot these new points. Connect them smoothly to the inverse graph.

Properties of Inverse Relations

Inverse relations are formed by swapping the x-value and y-value in each pair of a relation. When this is done, the domain becomes a range, and the range becomes a domain. Here are some properties given below:
 

• In an inverse relation, each ordered pair(x, y) becomes (y, x).
• The domain of the original relation becomes the range of the inverse, and vice versa.
• The graph of an inverse relation is a reflection of the original across the line y = x. 
• Doing the inverse of an inverse gives the original relation again.
• The inverse of a function may not be a function unless the real passes the horizontal line.
   

An inverse relation exchanges inputs and outputs, for example:

A={p, q, r, s, t}
B={1, 2, 3, 4, 5}

R={(p, 1}

Inverse Relation Theorem

In the inverse relation theorem, if you take the inverse of a relation, and then take the inverse again, you return to the original relation.
 

1. Start with a relation R, contains an ordered pair (x, y) 
2. Then, form the inverse relation R-1, (x, y) becomes (y, x)
3. Take the inverse (y, x)again, swap the elements that become (x, y)
4. So, the result that we get back to the original ordered pair (x, y)
5. This proves that (R-1)-1=R
    

Step 1: Choose points from the original graph. (0, 2).
 

Step 2: Swap the coordinates of the x- and y-values, so (0, 2) that it becomes (2, 0).
 

Step 3: Plot the new point. Do this for all original points.
 

Step 4: Connect the dots. The new graph is the inverse.

• (0, 2)(2, 0)
• (-2, 0)(0, -2)
• (-4, 2)(2, -4)
• (-2, 4)(4, -2)

Simply reflect the original points across the line y=x to get the inverse relation.

Understanding inverse relations is important because it helps in grasping how functions can be undone, how domains/ranges flip, and it appears in many contexts. Here are some important tips or tricks for students to master inverse relations. 

• Always swap the ordered pairs first: Whenever you have a relation given by pairs (𝑥,𝑦), immediately convert each into (𝑦,𝑥) for the inverse. This is the foundational step.
  
   
• Check domain: After forming the inverse, the original domain becomes the range and the original range becomes the domain. Recognizing this helps avoid mistakes.
  
   
• Graph by reflecting across the line 𝑦 = 𝑥: If you have a graph of the original relation (or function), plot points (𝑥,𝑦), then for the inverse plot (𝑦,𝑥). Visualizing this reflection helps deepen understanding.
  
   
• Be careful with functions vs relations: Not every relation has an inverse that is itself a function. A function’s inverse is a function only if it passes the horizontal-line test (i.e., the original was one-to-one).
  
   
• Use real-life contexts: Thinking about 'undoing' something helps. For example, if a function says “input → output,” then the inverse says “output → input”. The article gives examples like converting temperature scales or finding time from a growth formula.

We use real-life applications in many fields like cryptography, engineering, and finance. Here are some examples of real-life applications mentioned below:

 

• Setting the correct angle in medical image: Use by radiologists, medical technicians, who use inverse trigonometric functions (tan-1) to calculate the beam angle, if the depth and distance of a target are known. They use this to make sure CT/MRI scanners, especially for injuries and tumors.
  
   
• Time needed for population growth: Used by economists, public health experts, and demographers. To predict when a city's population will reach a milestone. Population over time: P(t)=P0ert. The inverse function solves for time t given a target population.
  
   
• Resolving work hours from pay checks: Use by hourly-paid workers and employers. When an employee sees his paycheck and rate, he can find out how many hours you worked by using the inverse function.
  
   
• From pH back to concentration: Use by chemists, and labs, it's essential to know acidity from pH measurements pH=-log[H+]; the inverse uses exponentials to find hydrogen ion concentration from pH.
  
   
• Temperature conversion: Use by students, travelers, and cooks. For Celsius to Fahrenheit is a function, while Fahrenheit to Celsius is its inverse. This lets you convert between two scales easily. For example, the inverse of the equation F=95 (C+32) is C=59(F-32).