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2. Inverse:The reversal of some process or operation. For functions, the reversal involves the interchange of the domain with the range. Along with the reversal of the domain and range there is a reversal of the operation that describes the function. A linear function converts Celsius temperature to Fahrenheit temperature. Lets look at two facts: Water freezes at 0ºC and at 32°F; Water boils at 100°C and at 212°F The graph of the function that converts Celsius temperature into Fahrenheit temperature is a line that passes through the points: (0, 32) and (100, 212).

3. Lets find the equation of this line. We already have the y-intercept (0, 32) We know through experience with this formula that is equally useful to know the inverse relationship. Converting Fahrenheit temperature to Celsius temperature. This inverse relationship is found by solving for C in terms of F in the Celsius to Fahrenheit equation:

4. * Solve for C subtract 32 from both sides * Multiply by the reciprocal of the coeff. Both functions are graphed above. Notice the interchange of the domain and range.

5. Note: It can easily be shown that if we start with a temperature given in Celsius, convert it to Fahrenheit, and then convert it back to Celsius, we arrive back at the starting point. This is the essence of the inverse function relationship: finding a function that reverses the process and brings you back to your starting point. Notation: The inverse of a function is denoted NOT ALL FUNCTIONS ARE REVERSIBLE! for example… This function does not have an inverse because there is now way of knowing if the number 4 in the range originated from x = 2 or x = -2.

6. The only functions that can be inverted are one-to-one functions. One-to-one Functions A function f is one-to-one if for all x 1 and x 2 in its domain, f(x 1 ) = f(x 2 ) implies that x 1 = x 2 Each one of the range elements corresponds to precisely one domain element.

7. Horizontal line test for functions: A function is one-to-one precisely when every horizontal line intersects its graph at most once. Ex 1: Show that

8. Knowing what the graphs look like we can get an idea of what numbers we can use to show it is or isn’t one-to-one. a.) the horizontal line y = 1 crosses the graph at (0, 1) and at (1, 1), so we can show that two different inputs give the same output.

9. b.) the graph implies that the function is one-to-one, but what if we graphed incorrectly? We must use the definition to show this. When a function is one-to-one we can find its inverse.

10. Properties of Inverse functions Suppose that f is a one-to-one function with domain X and range Y. Then…

11. Steps for finding inverses: 1. Given an equation, interchange x and y. 2. Solve for y in terms of x. 3. Replace y with f -1 (y). Ex 2: Show that the function defined by f(x) = 2x – 3 with the domain restricted to the set [-1, 4] is one-to-one and determine its inverse. Solution:The domain is restricted in this example to better see how the domains and ranges will interchange. one-to-one:

12. We must substitute the endpoints of the domain of f into the function to determine the range so that we know what the domain and range of the inverse function will be. f(-1) = 2(-1) – 3 = -5 f(4) = 2(4) – 3 = 5 We now know the domain and range of the inverse of f. inverse:1.) y = 2x – 3 x = 2y - 3 2.) x + 3 = 2y ½(x) + 3/2 = y

13. We can verify this relationship by noting: Graph: On board. The graph of the Inverse of a One-to-one Function The graph of y = f -1 (x) is the reflection of the graph of y = f(x) about the line y = x. p. 106 bottom

14. Ex 3:Sketch the graph of the inverse function of the function defined by Solution:This is a cubic function and we know the parent function is one-to-one, but we will have to check to see if this one is one-to-one. x y -1 -3 -0.5 -1.25 0 1 3 0.5 1.25 Plot these points and see if it passes the horizontal line test. We could find the inverse of this function, but it is not easily done.

15. We do know two facts that will aid us in sketching this graph: The domains and ranges interchange The graphs of inverse functions are reflections over the line y = x. Sketch on board. Ex 4:GivenDetermine the inverse of f if the domain is restricted to the interval [-2, ∞) Solution: We must complete the square so that our function has only one x.

16. If we graph f and restrict its domain we can determine the range: We also know that the domains and ranges interchange Now, lets graph…on board.