Finding the inverse of a function algebraically involves swapping the roles of $x$ and $y$ and then solving for $y$. This new equation represents the inverse function. The inverse function essentially "undoes" the original function. It's important to remember that not all functions have inverses, especially if they don't pass the horizontal line test. This worksheet will guide you through the process with examples tailored for Algebra 2.
Match the term with its definition:
| Term | Definition |
|---|---|
| 1. Inverse Function | A. A test to determine if a function has an inverse |
| 2. One-to-One Function | B. Swapping $x$ and $y$ and solving for $y$ |
| 3. Horizontal Line Test | C. A function that "undoes" the original function |
| 4. Finding Inverses Algebraically | D. A function where each $x$ value corresponds to a unique $y$ value |
| 5. Function | E. A relation where each input has exactly one output |
Answers: 1-C, 2-D, 3-A, 4-B, 5-E
To find the inverse of a function, first replace $f(x)$ with _____. Then, _____ the $x$ and $y$ variables. Finally, _____ for $y$ to get the inverse function, written as $f^{-1}(x)$. Remember that not all functions have inverses, especially if they don't pass the _____ line test. A function must be _____ to one to have an inverse.
Answers: $y$, swap, solve, horizontal, one
Explain why it's important to check if a function is one-to-one before finding its inverse. What happens if you try to find the inverse of a function that is not one-to-one? Give an example of such a function and its attempted inverse.
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