Algebra 2 Unit 9: Functional

Relationships

Topic: Functions & Their Inverses

Vocabulary Inverse Relation

– A relation that “undoes” a function.– The domain of a function is the range of its

inverse; the range of a function is the domain of its inverse.

– The graphs of a function & its inverse are symmetric about the line y = x.

One-to-one Function– A function in which each range value is

paired with one and only one domain value.– If a function, f is one-to-one, then it’s

inverse is also a one-to-one function and is notated f -1.

Determining whether a function is one-to-one

If a function passes the horizontal line test, it is a one-to-one function.– Any horizontal line must pass through the

graph of a function once and only once.

Function is one-to-one. Inverse will also be a function.

Function is not one-to one.

Inverse will not be a function.

Finding the inverse of a function

221 2)( xxf

221

221

2

2

yx

xy

Replace f (x) with y, then switch x & y in the equation.

42

)2(2

2

2

21

21

xy

yx

yx

yx

Solve the resulting equation for y.

The resulting relation is the inverse of f (x).

Take the square root of both sides (remember there are two solutions).

Subtract 2 from both sides.

Multiply both sides by 2.

Inverse Functions Determine if the given function is one-to-one. If

so, find its inverse & state its domain & range.

221 2)( xxf

The graph of the function does not pass the horizontal line test. It is not one-to-one, therefore its inverse is not a function (and we’re done with this problem!)

Inverse Functions Determine if the given function is one-to-one. If

so, find its inverse & state its domain & range.3 2)( xxf

The graph of the function does pass the horizontal line test. It is one-to-one, therefore its inverse is a function, and we must find it.

Inverse Functions Determine if the given function is one-to-one. If

so, find its inverse & state its domain & range.3 2)( xxf

3

3

2

2

yx

xy

Replace f (x) with y and switch x & y.

2

23

3

xy

yx

Solve for y to find the inverse.

2)( 31 xxf

Since we know this is a function, we must notate it properly (change y to f -1).

The domain & range of f (x) is all real #s, thus the domain & range of f -1(x) is all real #s.

Homework

Quest: Functions & Their Inverses

Due 5/7 (A-day) or 5/8 (B-day)