Section 1.5 Inverses of Functions. Box of Chocolates Section DSection ESection F EstherTamMonica.

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Presentation transcript:

Section 1.5 Inverses of Functions

Box of Chocolates Section DSection ESection F EstherTamMonica

The original functionThe 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. 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?

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

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.

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

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? Quick test - horizontal line test Note: invertible functions are called one-to-one functions. These are functions that are strictly increasing or strictly decreasing. Show algebraically that the composition ofwith is

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 andthen 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

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