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Inverse

Inverse
Inverse of an Operation

The quantity which cancels out the a given quantity. There are different kinds of inverses for different operations.

Table showing inverses: addition inverse of 3 is -3; multiplication inverse of 3 is 1/3; composition inverse of f(x)=2x-5 is...

 

See also

Additive inverse, multiplicative inverse, inverse function, inverse of a matrix

Key Formula

a+(−a)=0(additive inverse)a×1a=1,a≠0(multiplicative inverse)f(f−1(x))=x(inverse function)\begin{gathered}a + (-a) = 0 \qquad \text{(additive inverse)}\\a \times \frac{1}{a} = 1, \quad a \neq 0 \qquad \text{(multiplicative inverse)}\\f(f^{-1}(x)) = x \qquad \text{(inverse function)}\end{gathered}
Where:
  • aa = Any real number
  • −a-a = The additive inverse (opposite) of a
  • 1a\frac{1}{a} = The multiplicative inverse (reciprocal) of a, defined when a ≠ 0
  • f−1(x)f^{-1}(x) = The inverse function of f, which reverses the input-output relationship of f

Worked Example

Problem: Find the additive inverse and the multiplicative inverse of 5.
Step 1: To find the additive inverse, determine the number that adds with 5 to give 0.
5+?=05 + ? = 0
Step 2: The additive inverse of 5 is −5, because their sum is the additive identity (0).
5+(−5)=05 + (-5) = 0
Step 3: To find the multiplicative inverse, determine the number that multiplies with 5 to give 1.
5×?=15 \times ? = 1
Step 4: The multiplicative inverse of 5 is 1/5, because their product is the multiplicative identity (1).
5×15=15 \times \frac{1}{5} = 1
Answer: The additive inverse of 5 is −5, and the multiplicative inverse of 5 is 1/5.

Another Example

This example shows the inverse of a function rather than the inverse of a number, illustrating that the concept of 'inverse' extends beyond arithmetic to algebra and beyond.

Problem: Find the inverse function of f(x) = 3x + 6.
Step 1: Replace f(x) with y to write the equation in a simpler form.
y=3x+6y = 3x + 6
Step 2: Swap x and y. This reflects the idea that the inverse function reverses inputs and outputs.
x=3y+6x = 3y + 6
Step 3: Solve for y by subtracting 6 from both sides, then dividing by 3.
x−6=3y  ⟹  y=x−63x - 6 = 3y \implies y = \frac{x - 6}{3}
Step 4: Write the result as the inverse function and verify by composing f with its inverse.
f−1(x)=x−63f^{-1}(x) = \frac{x - 6}{3}
Step 5: Check: apply f to f⁻¹(x). You should get x back.
f ⁣(x−63)=3⋅x−63+6=(x−6)+6=x  ✓f\!\left(\frac{x-6}{3}\right) = 3\cdot\frac{x-6}{3} + 6 = (x-6)+6 = x \;\checkmark
Answer: The inverse function is f⁻¹(x) = (x − 6)/3.

Frequently Asked Questions

What is the difference between an additive inverse and a multiplicative inverse?
The additive inverse of a number a is −a, because a + (−a) = 0. The multiplicative inverse of a is 1/a (also called the reciprocal), because a × (1/a) = 1. Each one 'undoes' its respective operation and returns the identity element for that operation: 0 for addition, 1 for multiplication.
Does every number have an inverse?
Every real number has an additive inverse. For example, the additive inverse of 0 is 0 itself. However, only nonzero numbers have a multiplicative inverse, because 1/0 is undefined. Similarly, not every function has an inverse function—only one-to-one (injective) functions do.
What does 'inverse' mean in the context of functions?
An inverse function f⁻¹ reverses the action of f. If f takes an input a and produces output b, then f⁻¹ takes b and returns a. The key relationship is f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. Graphically, the inverse function is a reflection of the original over the line y = x.

Additive Inverse vs. Multiplicative Inverse

Additive InverseMultiplicative Inverse
DefinitionThe number that adds to a given number to produce 0The number that multiplies with a given number to produce 1
Formulaa + (−a) = 0a × (1/a) = 1, a ≠ 0
Other nameOppositeReciprocal
Example (a = 4)−41/4 or 0.25
Exists for zero?Yes: the additive inverse of 0 is 0No: 1/0 is undefined

Why It Matters

The concept of an inverse appears constantly throughout mathematics. You use additive inverses every time you solve an equation by subtracting from both sides, and multiplicative inverses every time you divide. In more advanced courses, inverse functions are essential for topics like logarithms (the inverse of exponentials), inverse trigonometric functions (arcsin, arccos, arctan), and inverse matrices used to solve systems of linear equations.

Common Mistakes

Mistake: Confusing the multiplicative inverse with the additive inverse. For example, thinking the 'inverse' of 3 is −3 when the question asks for the reciprocal.
Correction: Always identify which operation is involved. The additive inverse of 3 is −3 (opposite), while the multiplicative inverse of 3 is 1/3 (reciprocal). The word 'inverse' by itself requires context.
Mistake: Writing f⁻¹(x) as 1/f(x). Students sometimes interpret the −1 exponent as a reciprocal.
Correction: The notation f⁻¹(x) means the inverse function of f, not 1 divided by f(x). For instance, if f(x) = 2x, then f⁻¹(x) = x/2, which is not the same as 1/(2x).

Related Terms