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In this section, you will:
  • Verify inverse functions.
  • Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one.
  • Find or evaluate the inverse of a function.
  • Use the graph of a one-to-one function to graph its inverse function on the same axes.

A reversible heat pump is a climate-control system that is an air conditioner and a heater in a single device. Operated in one direction, it pumps heat out of a house to provide cooling. Operating in reverse, it pumps heat into the building from the outside, even in cool weather, to provide heating. As a heater, a heat pump is several times more efficient than conventional electrical resistance heating.

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. [link] provides a visual representation of this question. In this section, we will consider the reverse nature of functions.

Diagram of a function and would be its inverse.
Can a function “machine” operate in reverse?

Verifying that two functions are inverse functions

Suppose a fashion designer traveling to Milan for a fashion show wants to know what the temperature will be. He is not familiar with the Celsius scale. To get an idea of how temperature measurements are related, he asks his assistant, Betty, to convert 75 degrees Fahrenheit to degrees Celsius. She finds the formula

C = 5 9 ( F 32 )

and substitutes 75 for F to calculate

5 9 ( 75 32 ) 24 °C

Knowing that a comfortable 75 degrees Fahrenheit is about 24 degrees Celsius, he sends his assistant the week’s weather forecast from [link] for Milan, and asks her to convert all of the temperatures to degrees Fahrenheit.

A forecast of Monday’s through Thursday’s weather.

At first, Betty considers using the formula she has already found to complete the conversions. After all, she knows her algebra, and can easily solve the equation for F after substituting a value for C . For example, to convert 26 degrees Celsius, she could write

26 = 5 9 ( F 32 ) 26 9 5 = F 32 F = 26 9 5 + 32 79

After considering this option for a moment, however, she realizes that solving the equation for each of the temperatures will be awfully tedious. 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.

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.

Given a function f ( x ) , we represent its inverse as f 1 ( x ) , read as f inverse of x . The raised −1 is part of the notation. It is not an exponent; it does not imply a power of −1 . In other words, f 1 ( x ) does not mean 1 f ( x ) because 1 f ( x ) is the reciprocal of f and not the inverse.

The “exponent-like” notation comes from an analogy between function composition and multiplication: just as a 1 a = 1 (1 is the identity element for multiplication) for any nonzero number a , so f 1 f equals the identity function, that is,

Questions & Answers

what is the function of sine with respect of cosine , graphically
Karl Reply
tangent bruh
Aashish Reply
sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
sinx sin2x is linearly dependent
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
Aasik Reply
Wrong question
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
2x + 7 =19
2x +7=19. 2x=19 - 7 2x=12 x=6
because x is 6
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
Practice Key Terms 1

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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