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Inverse Operations and Functions

An operation we might do with a glove is put on. Another operation that could be done is take off. If we start with a bare hand:

Take a naked hand, put on a glove, and you get a hand with a glove. Then take off the glove and get a naked hand again.

If we start with a glove on:

Take a hand with a glove on, take of the glove, and you get a naked hand. Then put the glove bank on and get a hand with a glove on again.

The operations put on and take off undo each other. If we do one operation then the other, we end up where we started. Put on is the inverse operation to take offTake off is the inverse operation of put on.  Such operations form an operation-inverse operation pair.

The same is true in mathematics.  Most operations have an inverse operation. Starting with the simplest operations:

Take 79 add 256 to get 335 then subtract 256 and you get 79


Take 79 and subtract 246 to get -177 then add 256 to get 79

Add and Subtract are inverse operations. Similarly multiply and divide are inverse operations, except division by zero is not allowed.

Take negative 12 and multiple by negative 3 and you get 36 then divide by negative 3 and you get negative 12

Take negative 12 and divide by negative 3 and you get 4 then multiply by negative 3 and you get negative 12

You may have thought multiply and multiply by the reciprocal are the inverse pair. Since divide and multiply by the reciprocal are equivalent operations this is quite true.

Let's think about exponents.  We can get from a number to that number to the power of 2 by squaring the number.  To get back to the original number we need to take the square root.

-4 squared is 16 and the square root of 16 is -4

And in general, raising to a power and taking the root are inverse operations. Another common pair is cube-cube root.

five to the power n is 5^n and the nth root of 5^n is 5

Raising the base to a power and getting the logarithm (to that base) are also inverse operations.Recall that the expression y = 10x means y is equal to 10 raised to the power of xx is the exponent and 10 is the base.  This can also be written as x = log10 y.

32 = 2^ 5 take the log base 2 and get 5 . Raise 2 to this power and get 2^5 = 32

A pair that are very common from the various logarithms is the natural logarithm, ln, and the exponent, e.

take the natural log of x to get ln x the raise e to this power and get x


raise e to the power x and take the natural log and get x

The inverse trigonometric pairs are sin and sin-1, cos and cos-1 and tan with tan-1.These are dealt with in detail in Inverse Trigonometric Functions.

Sometimes an operation is its own inverse. Take a bus is an example.

Starting from home, take a bus to Massey then take a bus back home

A mathematical example is the reciprocal.

Take the reciprocal of 241/394 and get 394 divided by 241. Take the reciprocal of this and get 241/394


Evaluate each expression, and enter your answer as a number.

  1. (495 * 998) / 998 =
  2. the square root of 132 squared =
  3. 10 raised to log base 10 of 256 =
  4. the cube root of 8 squared =
  5. the natural log of e^3 =
  6. The reciprocal of 1/238 =
  7. log 10-4 =
  8. atan(tan 3) =

Match each of these operations with their inverse:

  1. The cubed root of x and
  2. sine and
  3. ex and
  4. arctangent and
  5. cosine and
  6. log x and

<< Order of Operations for Algebraic Expressions| Mathematical Formulae Index | Rearranging Equations I (Simple Equations) >>

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