Section 1.8 INVERSE FUNCTIONS

Function – is a relation that takes a value from set X and maps it into exactly one value from Set Y 1 2 3 4 5 10 8 6 1 2 2 4 3 6 4 8 10 5 Set X Set Y An inverse function would reverse that process, and map Set Y back into Set X

If we map what we get out of the function back, we will not always have a function going back. 1 2 2 4 3 6 4 8 5 In this example, going back, value 6 maps into both 3 and 5, therefore the reverse relation is NOT a function. Functions of this type are called many-to-one functions. Only functions that pair the y value (value in the range) with only one x will become functions going back the other way. These functions are called one-to-one functions.

This IS NOT a one-to-one function because y value is paired with more than one x value ( y is used more than once). 1 2 3 4 5 10 8 6 1 2 2 4 3 6 4 8 5 10 This IS a one-to-one function. Each x is paired with only one y and each y is paired with only one x. Only one-to-one functions will have inverse functions, meaning that mapping back to the original values will be also a function.

Function NOT a function Function Recall: to determine whether a relation is a function (given a graph), we can use a vertical line test. If a vertical line intersects the graph of an equation more than one time, the equation graphed is NOT a function. Function NOT a function Function

To be a one-to-one function, each y value could only be paired with one x. Let’s look at a couple of graphs. Look at a y value, Ex. y = 3. Is there only one corresponding x value? For any y value, Ex. y = - 1, y = 1, etc., how many corresponding x values do you see? This is a many-to-one function This IS a one-to-one function

If a horizontal line intersects the graph of an equation more than one time, the equation graphed is NOT a one-to-one function and will NOT have an inverse function. This is NOT a one-to-one, not a function This is NOT a one-to-one function This is a one-to-one function

These functions are reflections of each other about the line y = x Notice that the x and y values traded places for the function and its inverse. These functions are reflections of each other about the line y = x Let’s consider the function , compute some values, and graph them. This means “inverse function” x f (x) (2,8) -2 -8 -1 -1 0 0 1 1 2 8 (8,2) x f -1(x) -8 -2 -1 -1 0 0 1 1 8 2 Let’s take obtained values of the function and put them into the inverse function, then plot the points (-8,-2) (-2,-8) Yes, so it will have an inverse function Is this a one-to-one function? What will “undo” a cube? A cube root

Graphs of a function f and its inverse f -1 are reflections (symmetric) about the line y = x, and the x and y values have traded places. The domain of the function f is the range of the inverse f -1. The range of the function f is the domain of the inverse f -1. Algebraically, we can tell if two functions are inverses of each other if we substitute one into the other and that “undoes” the function. DEFINITION f and g are inverse functions if and only if their compositions for each x in the domain of g and for each x in the domain of f and

Verify that the functions f and g are inverses of each other. If we graph f(x) = (x – 2)2, it is a parabola shifted right 2. Is this a one-to-one function? This would not be one-to-one function unless we restrict the domain and only consider the function where x is greater than or equal to 2, so we have a one-to-one function.

are inverses of each other. Verify that the functions f and g are inverses of each other algebraically: Since both compositions are equal to x, according to the definition of inverse functions, f and g are inverses of each other.

confirm inverses by finding compositions Steps for Finding the Inverse of a One-to-One Function (the Algorithm) Use correct notation y = f -1(x) Solve for y And, confirm inverses by finding compositions Trade x and y places Replace f(x) with y

*Ensure f(x) is one-to-one first. Domain may need to be restricted. Confirm inverse by finding Find the inverse of Use correct notation y = f -1(x) or Solve for y Trade x and y places Replace f(x) with y *Ensure f(x) is one-to-one first. Domain may need to be restricted.