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### High School Mathematics8.5 Inverse Relation or Inverse Functions

 Inverse Relation: If R is a relation from set A into another set B, then by interchanging the first and second coordinates or ordered pairs of R we get a new relation. This relation is called the inverse relation of R and is denoted by R-|. Set builder form, R-| = {(y,x)/(x,y) Î R}. Note: Domain and range of R-| are respectively range and domin of R. Inverse Functions: To find the inverse of a function written as an equation, interchange the two variables x and y and solve for y. If the inverse is also a function, it is denoted by f-1 Example 1: Find the inverse of the function {(3,4),(5,6),(7,8),(9,10)}. State the domain and range of this inverse. State if the inverse is also a function. Original function: {(3,4),(5,6),(7,8),(9,10)} Interchange the first and second coordinates in each pair. Inverse of the function: {(4,3),(6,5),(8,7),(10,9)} domain: {4,6,8,10} range: {3,5,7,9} Since each element in the domain of the inverse maps onto one and only one element in the range, the inverse of the original function is also a function. Example 2: Find the inverse of the function y = 2x + 4. Original function: y = 2x + 4 Domain of f: {x: x is real} Range of f: {y: y is real} To find the inverse of the function interchange x and y x = 2y + 4 x/2 - 4 = 2y x/2 - 2 = y Therefore the inverse function f-1 is: y = x/2 - 2 Domain of f-1: {x: x is real} Range of f-1: {y: y is real} Directions: Choose the correct answer. Also write at least ten examples of your own.
 Q 1: If R = {(3,1),(5,2),(7,3),(9,4)} is a relation then find the domain of R-|.{3,5,7,9}{1,2,3,4} Q 2: If R = {(1,3),(2,5),(3,7),(4,9)} is a relation then find its inverse relation.{(1,3),(2,5),(3,7),(4,9)}{(3,1),(5,2),(7,3),(9,4)} Q 3: If R = {(1,2),(2,3),(3,4),(4,5)} is a relation then find its inverse relation.{(1,2),(2,3),(3,4),(4,5)}{(2,1),(3,2),(4,3),(5,4)} Q 4: If R = {(1,2),(2,3),(3,4),(4,5)} is a relation then find the domain of R-|.{1,2,3,4}{2,3,4,5} Q 5: If R = {(1,2),(3,4),(5,6)} is a relation then find the range of R-|.{2,4,6}{1,3,5} Q 6: If R-| = {(1,2),(3,4),(5,6)} is a relation then find its range.{2,4,6}{1,3,5} Q 7: If R-| = {(1,1),(2,2),(3,3),(4,4)} is a relation then find its range.{1,2,3,5}{1,2,3,4} Q 8: If R-| = {(1,3),(2,5),(3,7)} is a relation then find its range.{3,5,7}{1,2,3} Question 9: This question is available to subscribers only! Question 10: This question is available to subscribers only!

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