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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-| = {(1,1),(2,2),(3,3),(4,4)} is a relation then find its range.{1,2,3,4}{1,2,3,5} Q 2: Find the inverse of the function: y = 2x - 5, x = -1, -2, -3x = y/2 + 5/2; x = 7, 9, 11y = x + 5; x = 3, 5, 7y = x/2 + 5/2; x = -7, -9, -11 Q 3: Find the inverse of the function f(x) = -x + 7x = 7 - yx - y = 7y = 7- x Q 4: If R-| = {(1,2),(3,4),(5,6)} is a relation then find its range.{1,3,5}{2,4,6} Q 5: Find the inverse of the function: {(1,3),(2,3),(4,5),(9,5)}{(3,1),(3,2),(5,4),(5,9)}Not a function{(1,3),(2,3),(4,5),(9,5)} Q 6: If R-| = {(1,3),(2,5),(3,7)} is a relation then find its range.{3,5,7}{1,2,3} Q 7: Find the inverse of relation or function: y = 4x - 1x = 4/y - 1/yy = x/4 + 1/4x = 4y + 1 Q 8: 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)} Question 9: This question is available to subscribers only! Question 10: This question is available to subscribers only!