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3.6 Inverse Functions

3.6 Inverse Functions. Objectives: Define inverse relations & functions. Find inverse relations from tables, graphs, & equations. Determine whether an inverse relation is a function. Verify inverses using composition. Example #1 Graphing an Inverse Relation.

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3.6 Inverse Functions

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  1. 3.6 Inverse Functions Objectives: Define inverse relations & functions. Find inverse relations from tables, graphs, & equations. Determine whether an inverse relation is a function. Verify inverses using composition.

  2. Example #1Graphing an Inverse Relation • The graph of a function f is shown. Graph the inverse and describe the relationship between the function & its inverse. Make a table of points from the figure. Switch the x and y coordinates.

  3. Graphs of Inverse Relations The graph of the inverse f is a reflection of the graph of f across the line y = x. Graph the new set of points.

  4. Example #2Graphing an Inverse in Parametric Mode • Graph the function & its inverse in parametric mode.

  5. Example #3Finding an Inverse from an Equation • Find g(x), the inverse of This can be checked quickly by graphing the original and the inverse on the Y = screen. This does not need parametric mode.

  6. Example #3Finding an Inverse from an Equation • Find g(x), the inverse of To enter higher roots on the calculator, enter the root value first, then press MATH  5: x√

  7. Example #4Finding the Inverse from an Equation • Find the inverse of

  8. One-to-One Functions • A function is considered one-to-one if its inverse is also a function. • Use the horizontal line test to determine if the graph of the inverse will also be a function. • If the inverse is a function it is notated f -1. **This does not mean f to the -1 power.**

  9. Example #5Using the Horizontal Line Test • Graph each function below and determine whether it is one-to-one. Yes No Yes

  10. Example #6Restricting the Domain • Find an interval on which the function is one-to-one, and find f -1 on that interval. The function is one-to-one from [0, ∞). Using this domain the inverse would be the positive square root of x. Alternatively, if (−∞, 0] is chosen, negative square root of x is the inverse.

  11. Composition of Inverse Functions • A one-to-one function and its inverse have these properties. • Also, any two functions having both properties are one-to-one and inverses of each other. For every x in the domain of f and f -1

  12. Example #7Verifying the Inverse of a Function • Verify that f and g are inverses of each other. Since the both compositions of the functions equal x, then the functions are inverses of each other.

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