6.4 Inverse Functions6.4 Inverse FunctionsPart 1Part 1

Goal: Find inverses of linear Goal: Find inverses of linear functions.functions.

Inverse FunctionsInverse Functions

A function and its inverse can be described as A function and its inverse can be described as the "DO" and the "UNDO" functions. the "DO" and the "UNDO" functions. 

A function takes a starting value, performs some A function takes a starting value, performs some operation on this value, and creates an output operation on this value, and creates an output answer.  answer. 

The inverse of this function takes the output The inverse of this function takes the output answer, performs some operation on it, and answer, performs some operation on it, and arrives back at the original function's starting arrives back at the original function's starting value. value.

Definition of an InverseDefinition of an Inverse

  The The inverseinverse of a functionof a function is the set of is the set of ordered pairs obtained by ordered pairs obtained by interchanging interchanging the first and second elements of each pair the first and second elements of each pair in the original function. in the original function.               

Notation:Notation:   If    If  f f (x) is a given function, then  (x) is a given function, then

ff -1-1(x) denotes the inverse of(x) denotes the inverse of  f  f..

Finding the Inverse of a FunctionFinding the Inverse of a Function

Basically, the process of finding an inverse Basically, the process of finding an inverse is simply the swapping of the is simply the swapping of the xx and and yy coordinates.  coordinates. 

This newly formed inverse will be a This newly formed inverse will be a relation, but may relation, but may notnot necessarily be a necessarily be a function.  function. 

Horizontal Line TestHorizontal Line Test

The inverse of a function f is also a function The inverse of a function f is also a function if and only if no horizontal line intersects the if and only if no horizontal line intersects the graph of f more that once.graph of f more that once.

Three Methods to find the Inverse:Three Methods to find the Inverse:

1.1. Reflect graph over the line y = x.Reflect graph over the line y = x.

2.2. Swapping x and y-valuesSwapping x and y-values

3.3. Solving Algebraically:Solving Algebraically:a. Set the function = a. Set the function = yyb.  Swap the b.  Swap the xx and and yy variables variablesc.  Solve for c.  Solve for yy

Reflect the graph over y = xReflect the graph over y = x

Graph original function Graph original function

f(x) = 2f(x) = 2xx + 3 + 3It is drawn in It is drawn in blueblue. .

If reflected over the identityIf reflected over the identity

line, line, y = xy = x, the original , the original

function becomes the function becomes the redred

dotted graph. dotted graph. 

Swapping x and y-valuesSwapping x and y-values

Given relation, find the inverse Given relation, find the inverse relation.relation.

xx -2-2 -1-1 00 11 22

yy 44 22 00 -2-2 -4-4

xx 44 22 00 -2-2 -4-4

yy -2-2 -1-1 00 11 22

Solving AlgebraicallySolving Algebraically

Find the equation of the inverse of the Find the equation of the inverse of the relation f(x) = 2x – 4.relation f(x) = 2x – 4.

1.1. Set the function = Set the function = yy

y = 2x – 4 y = 2x – 4

2.2. Swap the Swap the xx and and yy variables variables

x = 2y -4x = 2y -4

3.3. Solve for Solve for yy

y = (x + 4)/2y = (x + 4)/2

Find the equation for the Inverse Find the equation for the Inverse function. function.

f(x) = -2x + 5f(x) = -2x + 5

ff-1-1(x) = -(x – 5)/2(x) = -(x – 5)/2

Verifying Inverse FunctionsVerifying Inverse Functions

Verify that f(x) = 2x – 4 and fVerify that f(x) = 2x – 4 and f-1-1(x) = ½x + 2 (x) = ½x + 2 are inverses.are inverses.

6.3 Composition 6.3 Composition Functions Functions

(Inverse Functions)(Inverse Functions)Part 2Part 2

Goal: Evaluate composition Goal: Evaluate composition functions and prove functions are functions and prove functions are

inverses of each other using inverses of each other using composition functions.composition functions.

Composition of FunctionsComposition of Functions

The composition of a function g with a The composition of a function g with a function f is:function f is:

h(x) = g(f(x))h(x) = g(f(x))

Example 1Example 1

Let f(x) = 3x – 14 and g(x) = x + 5. What is Let f(x) = 3x – 14 and g(x) = x + 5. What is the value of g(f(4))?the value of g(f(4))?

•First find f(4).First find f(4).

f(x) = 3x – 14 f(x) = 3x – 14

f(4) = 3(4) – 14 = -2f(4) = 3(4) – 14 = -2

•Then find g(-2).Then find g(-2).

g(x) = x + 5g(x) = x + 5

g(-2) = -2 + 5 = 3g(-2) = -2 + 5 = 3

Example 2Example 2

Let f(x) = 3x + 2 and g(x) = 2x – 7. Find Let f(x) = 3x + 2 and g(x) = 2x – 7. Find f(g(x)).f(g(x)).

•Find g(x) first.Find g(x) first.

g(x) = 2x – 7g(x) = 2x – 7

•Then find f(2x – 7)Then find f(2x – 7)

f(x) = 3x + 2f(x) = 3x + 2

f(2x – 7) = 3(2x – 7) + 2f(2x – 7) = 3(2x – 7) + 2

f(2x – 7) = 6x – 21 + 2f(2x – 7) = 6x – 21 + 2

f(2x – 7) = 6x – 19f(2x – 7) = 6x – 19

Therefore, f(g(x)) = 6x – 19 Therefore, f(g(x)) = 6x – 19

This "DO" and "UNDO" process can be This "DO" and "UNDO" process can be stated as a composition of functions. stated as a composition of functions.

If functions If functions ff and and gg are inverse functions, are inverse functions, f(g(x)) = g(f(x)) = x.f(g(x)) = g(f(x)) = x.

Example: If f(x) = x-1 and g(x) = x +1Example: If f(x) = x-1 and g(x) = x +1

then f(g(x)) = x and g(f(x)) = xthen f(g(x)) = x and g(f(x)) = x

Think of them as "undoing" one another Think of them as "undoing" one another and leaving you right where you started. and leaving you right where you started.

Verify that f and g are inverse Verify that f and g are inverse functions. functions.

f(x) = 3x – 1 and g(x) = ⅓x + ⅓f(x) = 3x – 1 and g(x) = ⅓x + ⅓