Algebra 2 Notes
Algebra 2 Notes Name: ________________
Functions and Their Inverses
DAY ONE:
you learned that the inverse of a function [pic] “undoes” [pic]. Its graph is a reflection across the line [pic]. The inverse may or may not be a function.
Recall that the ___________________- line test can help you determine whether a relation is a function. Similarly, the ___________________- line test can help you determine whether the inverse of a function is a function.
|The Horizontal Line Test |
|Words |Examples |
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| |[pic] |[pic] |
|If any horizontal line passes through more than | | |
|one point on the graph of a relation, the inverse | | |
|relation is not a function. |The inverse __________ a function. |The inverse __________ a function. |
Example 1: Use the horizontal-line test to determine whether the inverse of each relation is a function.
|a. |b. |c. |d. |
|[pic] |[pic] |[pic] |[pic] |
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Recall from Section 7.2 that to write the rule for the inverse of a function, you can exchange [pic] and [pic]. Because the values of [pic] and [pic] are switched, the domain of the function will be the range of its inverse and vice versa.
Example 2: Find the inverse of each function. Determine whether it is a function, and state its domain and its range.
|a. [pic] |b. [pic] |
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DAY TWO:
You have seen that the inverses of functions are not necessarily functions. When both a relation and its inverse are functions, the relation is called a _________________________________________. In a one-to-one function, each [pic]-value is paired with exactly one [pic]-value.
You can use composition of functions to verify that two functions are inverses. Because inverse functions “undo” each other, when you compare two inverses the result is the input ________.
|Identifying Inverse Functions |
|Words |Algebra |Example |
|If the compositions of two functions equal the |If [pic], then [pic] and [pic] are inverse |[pic] and [pic] |
|input value, the functions are inverses. |functions. |[pic] |
| | |[pic] |
Example 3: Determine by composition whether each pair of functions are inverses or not.
|a. [pic] and [pic] |b. For [pic], [pic] and [pic]. |
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