Inverse Relations and Functions

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Inverse Relations If a relation maps element If a relation maps element a a of its domain to of its domain to

element element bb of its range, the INVERSE of its range, the INVERSE RELATION “undoes” the relation and maps RELATION “undoes” the relation and maps bb back to back to aa. So if (a,b) is an ordered pair of a . So if (a,b) is an ordered pair of a relation, then (b,a) is an ordered pair of its relation, then (b,a) is an ordered pair of its inverse.inverse.

1.21.21.41.41.61.61.91.9

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1.21.21.41.41.61.61.91.9

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RELATION RELATION rr INVERSE OF RELATION INVERSE OF RELATION rr

Interchanging x and y

Find the inverse of y = x2 + 3 Find the inverse of y = x2 + 3

yx

yx

yx

xy

3

3

3

3

2

2

2

yx

yx

yx

xy

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3

3

3

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2

Functions and Relations

Does define a function? Is its inverse a function?

32 xy

3

32

xy

xy

3

32

xy

xy

Graphing a Relation and its Inverse The reflection over the line y = x is the graph

of the inverse. You can also find points on the graph of the

inverse by reversing the coordinates of the points on the relation.

Notation

The inverse of f is denoted by f -1. Read f -1 as “the inverse of f” or as “f inverse.”

The notation f(x) is used for functions, but f -1(x) may be a relation that is not a function.

If f and f -1, are functions, they are called inverse functions. For inverse functions

and xxff 1 xxff 1

Real Life Problems with Formulas Functions that model real-life situations are

frequently expressed as formulas with letters that remind you of the variables they represent. When finding the inverse of a formula, it would be very confusing to interchange the letters. Keep the letters the same and just solve the formula for the other variable