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Section 1.5 Inverses of Functions

Section 1.5 Inverses of Functions. Box of Chocolates. The original function. The inverse of the original function. Example1. Given . Write the equation for the inverse relation by interchanging the variables and transforming the equation so that y is in terms of x.

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Section 1.5 Inverses of Functions

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  1. Section 1.5 Inverses of Functions

  2. Box of Chocolates

  3. The original function The inverse of the original function

  4. Example1 Given Write the equation for the inverse relation by interchanging the variables and transforming the equation so that y is in terms of x. Plot the function and its inverse on the same screen, using equal scales for the two axes. Explain why the inverse relation is not a function. Plot the line y = x on the same screen. How are the graph of the function and its inverse relation related to this line?

  5. Parametric Equations Using a third variable to relate x and y One value for the parameter yields values for both x and y This third variable is called the "parameter." On your calculator, it's "t".

  6. Plotting inverse images becomes particularly easy using parametric equations. If set x equal to t, we can simply rewrite f(x) in terms of t and reproduce our original function as a pair of parametric equations. Since the first step we take in determining an inverse relationship is switching our x and y variables in our function expression, we can obtain the same result by switching the x-t and y-t relationships in our parametric equations.

  7. Example 2 Plot the graph of for x in the domain and its inverse using parametric equations (Domain and range of both?)

  8. Example 3 Let Find the equation of the inverse of f. Plot function f and its inverse on the same screen. Is f an invertible function? Why or why not? Show algebraically that the composition of with is Note: invertible functions are called one-to-one functions. These are functions that are strictly increasing or strictly decreasing. Quick test - horizontal line test

  9. Definitions and Properties: Function Inverses The inverse of a relation in two variables is formed by interchanging the two variables. If the inverse of function f is also a function, then f is invertible. If f is invertible and then you can write the inverse of f as To plot the graph of the inverse of a function, either Interchange the variables, solve for y, and plot the resulting equation(s), Or Use parametric mode

  10. If f is invertible, then the compositions of and are provided x is in the domain of f and is in the domain of provided x is in the domain of and is in the domain of A one-to-one function is invertible. Strictly increasing or strictly decreasing functions are one-to-one functions.

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