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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:

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:

and

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

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.

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

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.

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

and

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.

A mathematical example is the reciprocal.

### Exercise

1. (495 * 998) / 998 =
2. =
3. =
4. =
5. =
6. =
7. log 10-4 =
8. atan(tan 3) =

Match each of these operations with their inverse:

1. and
2. sine and
3. ex and
4. arctangent and
5. cosine and
6. log x and