1.6 Inverse Functions

Objectives Find inverse functions informally and verify that two functions are inverse functions of each other. Determine from the graphs of functions whether the functions have inverse functions. Determine if functions are one to one. Find inverse functions algebraically.

Mapping Relations Mapping shows how each member of the domain and range are paired –Example: –1. Relation: {(1,2), (-3,9), (7,-5)} –2. Relation: {(4,-6), (1,3), (1,0), (-2,7)}

Mapping Example Relation  {(1, 0), (5, 2), (7, 2), (-1, 11)} Domain  {1, 5, 7, -1} Range  {0, 2, 11}

Types of Relations

Which Relations are also Functions? Many to One Relationship One to One Relationship { (3, 2), (1, 2), (2, 2), (8, 2), (7, 2) } { (0, 2), (1, 0), (2, 6), (8, 12) }

What is an Inverse Function?  INVERSE FUNCTION – reversing a function, “undoing” it.  f -1 notates an inverse function. (not 1/f)

Set X Set Y Remember we talked about functions---taking a set X and mapping into a Set Y An inverse function would reverse that process and map from SetY back into Set X

DomainRange Inverse relation x = |y| x y Domain Range x y Function y = |x| + 1 Every function y = f (x) has an inverse relation x = f (y). The ordered pairs of : y = |x| + 1 are {(-2, 3), (-1, 2), (0, 1), (1, 2), (2, 3)}. x = |y| + 1 are {(3, -2), (2, -1), (1, 0), (2, 1), (3, 2)}. The inverse relation is not a function. It pairs 2 to both -1 and +1. Inverse Relation

Inverse Relations: Ordered Pairs Original relation, y 1 : {(0, 4) (2, 6) (5, 9) (10, 14)} Inverse relation, y 2 : {(4, 0) (6, 2) (9, 5) (14, 10)} To find the inverse relation, represented by ordered pairs, simply switch the x and y of each ordered pair. If y 1 contains (x, y), then y 2 contains (y, x). Are these two relations, a function (f(x)) and it’s inverse function (f -1 (x))? YES

If we map what we get out of the function back, we won’t always have a function going back. Since going back, 6 goes back to both 3 and 5, the mapping going back is NOT a function These functions are called many-to-one functions Only functions that pair the y value (value in the range) with only one x will be functions going back the other way. These functions are called one-to-one functions.

A function y = f (x) with domain D is one-to-one on D if and only if for every x 1 and x 2 in D, f (x 1 ) = f (x 2 ) implies that x 1 = x 2. A function is a mapping from its domain to its range so that each element, x, of the domain is mapped to one, and only one, element, f (x), of the range. A function is one-to-one if each element f (x) of the range is mapped from one, and only one, element, x, of the domain. One-to-One Functions

This would not be a one-to-one function because to be one-to-one, each y would only be used once with an x. This function IS one-to-one. 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 the mapping back to the original values is also a function.

Recall that to determine by the graph if an equation is a function, we have the vertical line test. If a vertical line intersects the graph of an equation more than one time, the equation graphed is NOT a function. This is a function This is NOT a function This is a function

x y 2 2 Horizontal Line Test A function y = f (x) is one-to-one if and only if no horizontal line intersects the graph of y = f (x) in more than one point. y = 7 Example: The function y = x 2 – 4x + 7 is not one-to-one on the real numbers because the line y = 7 intersects the graph at both (0, 7) and (4, 7). (0, 7) (4, 7)

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 (for example y = 3)and see if there is only one x value on the graph for it. This is a many-to-one function For any y value, a horizontal line will only intersection the graph once so will only have one x value This then 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 a one-to-one function This is NOT a one-to- one function

one-to-one Example: Apply the horizontal line test to the graphs below to determine if the functions are one-to-one. a) y = x 3 b) y = x 3 + 3x 2 – x – 1 not one-to-one x y x y 4 4 8

Why are one-to-one functions important? One-to-One Functions have Inverse functions

A function, f, has an inverse function, g, if and only if (iff) the function f is a one-to-one (1-1) function. Existence of an Inverse Function

The ordered pairs of the function f are reversed to produce the ordered pairs of the inverse relation. Example: Given the function f = {(1, 1), (2, 3), (3, 1), (4, 2)}, its domain is {1, 2, 3, 4} and its range is {1, 2, 3}. The inverse relation of f is {(1, 1), (3, 2), (1, 3), (2, 4)}. The domain of the inverse relation is the range of the original function. The range of the inverse relation is the domain of the original function. Ordered Pairs

Domain of fRange of f

Domain and Range The domain of f is the range of f -1 The range of f is the domain of f -1 Thus... we may be required to restrict the domain of f so that f -1 is a function

Restricting a Domain When the inverse of a function is not a function, the domain of the function can be restricted to allow the inverse to be a function. In such cases, it is convenient to consider “part” of the function by restricting the domain of f(x). If the domain is restricted, then its inverse is a function.

Restricting the Domain Recall that if a function is not one-to-one, then its inverse will not be a function.

Restricting the Domain If we restrict the domain values of f(x) to those greater than or equal to zero, we see that f(x) is now one-to-one and its inverse is now a function.

Your Turn: (c)Suggest a suitable domain for so that the inverse function can be found. (a)Sketch the function where. (d)On the same axes sketch. (b)Write down the range of.

(a) Solution: ( We’ll look at the other possibility next. ) (c) Restricted domain: (b) Range of : (c)Suggest a suitable domain for so that the inverse function can be found. (a)Sketch the function where. (d)On the same axes sketch. (b)Write down the range of.

The other possibility: (a) (c) Suppose you chose for the domain (b) Range of : (c)Suggest a suitable domain for so that the inverse function can be found. (a)Sketch the function where. (d)On the same axes sketch. (b)Write down the range of.

y = x The graphs of a relation and its inverse are reflections in the line y = x. The ordered pairs of f are given by the equation. Example: Find the graph of the inverse relation geometrically from the graph of f (x) = x y The ordered pairs of the inverse are given by.

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

Inverses of Functions If the inverse of a function f is also a function, it is named f  1 and read “f- inverse.” The negative 1 in f  1 is not an exponent. This does not mean the reciprocal of f. f  1 (x) is not equal to

y = f(x) y = x y = f -1 (x) Example: From the graph of the function y = f (x), determine if the inverse relation is a function and, if it is, sketch its graph. The graph of f passes the horizontal line test. The inverse relation is a function. Reflect the graph of f in the line y = x to produce the graph of f -1. x y Determine Inverse Function

Consider the graph of the function The inverse function is Example:

Consider the graph of the function The inverse function is An inverse function is just a rearrangement with x and y swapped. So the graphs just swap x and y ! x x x x

x x x x is a reflection of in the line y = x What else do you notice about the graphs? x The function and its inverse must meet on y = x

Your Turn: Graph f(x) and f -1 (x) on the same graph.

On the same axes, sketch the graph of and its inverse. x Solution: Your Turn:

Example: Find the inverse relation algebraically for the function f (x) = 3x + 2. y = 3x + 2 Original equation defining f x = 3y + 2 Switch x and y. 3y + 2 = x Reverse sides of the equation. To calculate a value for the inverse of f, subtract 2, then divide by 3. y = Solve for y. To find the inverse of a relation algebraically, interchange x and y and solve for y. Inverse Relation Algebraically

For a function y = f (x), the inverse relation of f is a function if and only if f is one-to-one. For a function y = f (x), the inverse relation of f is a function if and only if the graph of f passes the horizontal line test. If f is one-to-one, the inverse relation of f is a function called the inverse function of f. The inverse function of y = f (x) is written y = f -1 (x). Inverse Function

Steps for Finding the Inverse of a One-to-One Function Replace f(x) with y Trade x and y places Solve for yy = f -1 (x)

Find the inverse of Replace f(x) with y Trade x and y places Solve for yy = f -1 (x) Let’s check this by doing or Ensure f(x) is one to one first. Domain may need to be restricted.

Your Turn: Find the inverses of these functions: 1) 2) 3)4) 5)6)

So geometrically if a function and its inverse are graphed, they are reflections about the line y = x and the x and y values have traded places. The domain of the function is the range of the inverse. The range of the function is the domain of the inverse. Also if we start with an x and put it in the function and put the result in the inverse function, we are back where we started from. Given two functions, we can then tell if they are inverses of each other if we plug one into the other and it “undoes” the function. Remember subbing one function in the other was the composition function. So if f and g are inverse functions, their composition would simply give x back. For inverse functions then:

A function, f, has an inverse function, g, if and only if f(g(x)) = x and g(f(x)) = x, for every x in domain of g and in the domain of f. Alternate Definition of an Inverse Function

The inverse function is an “inverse” with respect to the operation of composition of functions. The inverse function “undoes” the function, that is, f -1 ( f (x)) = x. The function is the inverse of its inverse function, that is, f ( f -1 (x)) = x. Example: The inverse of f (x) = x 3 is f -1 (x)=. 3 f -1 ( f(x))= = x and f ( f -1 (x)) =( ) 3 = x Composition of Functions

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

Verify that the functions f and g are inverses of each other. Since both of these = x, if you start with x and apply the functions they “undo” each other and are inverses.

It follows that g = f -1. Example: Verify that the function g(x) = is the inverse of f(x) = 2x – 1. f(g(x)) = 2g(x) – 1 = 2( ) – 1 = (x + 1) – 1 = x g( f(x)) = = = = x

Homework Section 1.6, pg. 69 – 72: Vocabulary Check #1 – 5 all Exercises: #1-33 odd, odd, 111, 113