Inverse Functions and their Representations Lesson 5.2

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 = TempR = Resistance R = f(T)

Example We could also take the view that we wish to determine T, temperature as a function of R, resistance. R = ResistanceT = Temp 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? NO … when we reverse the ordered pairs, the result is Not a function We would say the function is not one-to-one A function is one-to-one when different inputs always result in different outputs xY

One-to-One Functions When different inputs produce the same output Then an inverse of the function does not exist When different inputs produce different outputs Then the function is said to be “one-to-one” Every one-to-one function has an inverse Contrast

One-to-One Functions Examples Horizontal line test?

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) = x Determine the inverse function Problem => f -1 (x) is not a function

Assignment Lesson 5.2 Page 396 Exercises 1 – 93 EOO