1. What the Inverse of a Function Means
The inverse of a function reverses the original mapping. If a function sends x to y, the inverse sends y back to x. It undoes the action of the function.
So, an inverse function answers the question: “Which input produced this output?”
2. Formal Definition
If f is a function from A to B, its inverse f−1 is a function from B back to A, provided the inverse exists.
2.1. Definition
f(a) = b \quad \Rightarrow \quad f^{-1}(b) = a
The inverse reverses each pair of the original function.
3. When Does an Inverse Exist?
A function has an inverse only when it matches outputs uniquely and covers the entire codomain. In other words, the function must be bijective.
3.1. Condition
f \text{ has an inverse } \iff f \text{ is bijective}
4. How to Find the Inverse from a Formula
To find the inverse of a function given by a rule, follow three basic steps:
4.1. Steps
- Start with y = f(x).
- Solve this equation for x in terms of y.
- Swap x and y to get the inverse function.
4.2. Example
Let f(x) = 3x + 2.
y = 3x + 2
3x = y - 2
x = \frac{y - 2}{3}
f^{-1}(x) = \frac{x - 2}{3}
5. Inverse Using Ordered Pairs
When a function is given as a set of ordered pairs, the inverse is obtained by swapping each pair.
5.1. Example
f = \{(1,4),(2,5),(3,6)\}
Swap each pair:
f^{-1} = \{(4,1),(5,2),(6,3)\}
6. Graphical Meaning of the Inverse
The graph of the inverse function is the reflection of the graph of the original function across the line y = x. Every point (a, b) on the original graph appears as (b, a) on the inverse graph.
6.1. Horizontal Line Test
A function has an inverse if every horizontal line intersects its graph at most once.
7. Inverse Does Not Always Exist
Functions that repeat output values cannot have an inverse because you cannot reverse a mapping that merges inputs.
7.1. Example
f(x) = x^2
f(2) = 4 and f(-2) = 4 → you cannot decide whether 4 came from 2 or -2.
However, restricting the domain (e.g., x ≥ 0) makes the inverse possible.
8. Notation and Interpretation
The inverse is written as f−1, but this does not mean 1/f. It is the function that reverses f.
f^{-1}(f(x)) = x
f(f^{-1}(x)) = x
9. Why Inverse Functions Matter
Inverse functions help solve equations, reverse transformations, and understand relationships between variables. They are essential in algebra, logarithms, trigonometry, and calculus.