Algebra 2 Unit 9: Functional Relationships Topic: Functions & Their Inverses.
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Transcript of Algebra 2 Unit 9: Functional Relationships Topic: Functions & Their Inverses.
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)