• MathsFirst Home
• Online Maths Help
  • Notation
  • Arithmetic
    • Decimals
    • Fractions
    • HCF and LCM
    • Order of Operations
    • Signed Numbers
  • Algebra
    • Exponents
      • Integer Exponents
      • Fractional Exponents
      • Exponents Worksheet
    • Combining Like Terms
    • Simple Expansions
    • Factorisation
    • Mathematical Formulae
      • Constructing Formulae
      • Rearranging Formulae
        • Order of Operations for Algebraic Expressions
        • Inverse Operations and Functions
        • Rearranging Equations I (Simple Equations)
        • Rearranging Equations II (Quadratic Equations)
        • Rearranging Equations III (Harder Examples)
      • Area, Surface Area and Volume Formulae
    • Coordinate Systems and Graphs
    • Linear Equations & Graphs
    • Simultaneous Linear Equations
    • Polynomials
    • Quadratic Polynomials
    • Rational Functions
    • Logarithms
  • Calculus
    • Differentiation
      • Tangents, Derivatives and Differentiation
      • The Basic Rules
      • Products and Quotients
      • Chain Rule
      • Logarithmic Differentiation
      • Mixed Differentiation Problems
      • Sign of the Derivative
      • Sign of the Second Derivative
    • Integration
      • Basic Integrals
      • Multiple, Sum and Difference Rules
      • Linear Substitution
      • Simpler Integration by Substitution
      • Harder Integration by Substitution
      • Trig Substitution 1
      • Trig Substitution 2
      • Integration by Parts
  • Trigonometry
    • Pythagoras Theorem
    • Trig Functions and Related Topics
• First Year Maths Pathways
• Readiness Quizzes
• Study Maths
  at Massey
  • Undergraduate
  • Postgraduate
• Other Math Links
• About MathsFirst
• Contact Us

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:

If we start with a glove on:

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 off.  Take 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 = 10^x means y is equal to 10 raised to the power of x.  x 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.

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