High School Mathematics 8.5 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.