Inverse Functions Lesson 8.2

Definition A function is a set of ordered pairs with no two first elements alike. f(x) = { (x,y) : (3, 2), (1, 4), (7, 6), (9,12) } But ... what if we reverse the order of the pairs? This is also a function ... it is the inverse function f -1(x) = { (x,y) : (2, 3), (4, 1), (6, 7), (12, 9) }

Example Consider an element of an electrical circuit which increases its resistance as a function of temperature. T = Temp R = Resistance -20 50 150 20 250 40 350 R = f(T)

Now we would say that g(R) and f(T) are inverse functions Example We could also take the view that we wish to determine T, temperature as a function of R, resistance. R = Resistance T = Temp 50 -20 150 250 20 350 40 T = g(R) Now we would say that g(R) and f(T) are inverse functions

Terminology If R = f(T) ... resistance is a function of temperature, Then T = f-1(R) ... temperature is the inverse function of resistance. f-1(R) is read "f-inverse of R“ is not an exponent it does not mean reciprocal

Does This Have An Inverse? Given the function at the right Can it have an inverse? Why or Why Not? x Y 1 5 2 9 4 6 7 NO … when we reverse the ordered pairs, the result is Not a function.

Finding the Inverse Try

Composition of Inverse Functions Consider f(3) = 27   and   f -1(27) = 3 Thus, f(f -1(27)) = 27 and f -1(f(3)) = 3 In general   f(f -1(n)) = n   and f -1(f(n)) = n (assuming both f and f -1 are defined for n)

Graphs of Inverses Again, consider Set your calculator for the functions shown Dotted style Use Standard Zoom Then use Square Zoom

Graphs of Inverses Note the two graphs are symmetric about the line y = x

Investigating Inverse Functions Consider Demonstrate that these are inverse functions What happens with   f(g(x))? What happens with  g(f(x))? Define these functions on your calculator and try them out

Domain and Range The domain of f is the range of f -1 The range of f is the domain of f -1 Thus ... we may be required to restrict the domain of f so that f -1 is a function

Domain and Range Consider the function h(x) = x2 - 9 Determine the inverse function Problem =>  f -1(x) is not a function

Inverse Pumpkins In a recent pumpkin launching contest, one launcher misfired so that the pumpkin went straight up into the air (!!) and came back down to land on the launch personnel!   Below is the graph of the height of the pumpkin as a function of time h(t)  Note ... the curve is not the path of the pumpkin. 

Inverse Pumpkins What is the "hang time" of the launch? Restrict the domain of h(t) so that it has an inverse that is a function Graph the inverse of this function Change the story to go with your new graph.  Explain in your story why it makes sense that the inverse is a function.

Assignment Lesson 8.2 Page 370 Exercises 1 – 49 EOO