The “exponent-like” notation comes from an analogy between function composition and multiplication: just as a − 1 a = 1 a − 1 a = 1 (1 is the identity element for multiplication) for any nonzero number a, a, so f − 1 ∘ f f − 1 ∘ f equals the identity function, that is,įigure 3 Domain and range of a function and its inverse In other words, f − 1 ( x ) f − 1 ( x ) does not mean 1 f ( x ) 1 f ( x ) because 1 f ( x ) 1 f ( x ) is the reciprocal of f f and not the inverse.

It is not an exponent it does not imply a power of −1 −1. ” The raised −1 −1 is part of the notation. Given a function f ( x ), f ( x ), we represent its inverse as f − 1 ( x ), f − 1 ( x ), read as “ f “ f inverse of x. The formula for which Betty is searching corresponds to the idea of an inverse function, which is a function for which the input of the original function becomes the output of the inverse function and the output of the original function becomes the input of the inverse function. She realizes that since evaluation is easier than solving, it would be much more convenient to have a different formula, one that takes the Celsius temperature and outputs the Fahrenheit temperature. In this section, we will consider the reverse nature of functions.Ģ6 = 5 9 ( F − 32 ) 26 ⋅ 9 5 = F − 32 F = 26 ⋅ 9 5 + 32 ≈ 79 26 = 5 9 ( F − 32 ) 26 ⋅ 9 5 = F − 32 F = 26 ⋅ 9 5 + 32 ≈ 79Īfter considering this option for a moment, however, she realizes that solving the equation for each of the temperatures will be awfully tedious. Figure 1 provides a visual representation of this question. If some physical machines can run in two directions, we might ask whether some of the function “machines” we have been studying can also run backwards.

As a heater, a heat pump is several times more efficient than conventional electrical resistance heating. Operating in reverse, it pumps heat into the building from the outside, even in cool weather, to provide heating. Operated in one direction, it pumps heat out of a house to provide cooling. Use the graph of a one-to-one function to graph its inverse function on the same axes.Ī reversible heat pump is a climate-control system that is an air conditioner and a heater in a single device.Find or evaluate the inverse of a function.Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one.

Inverse graph