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URL: https://nakafa.com/en/subjects/mathematics/function-composition-inverse-function/inverse-function
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/function-composition-inverse-function/inverse-function/en.mdx
Learn inverse functions one step at a time: definition, bijective requirements, finding formulas, and graph reflections. Learn f⁻¹(x) with clear examples.
---
## Understanding Inverse Functions
You often translate words or sentences from English to Indonesian, for example, when watching movies or reading news. This translation process is similar to how a function works: there is an input (English word) and an output (Indonesian word).
Consider the following illustration:
Component: FunctionIllustration
Props:
- title: Translation Machine
- description: The translation machine changes words or sentences from English to Indonesian.
- machineLabel: Translation Machine
- content: { input: "Mathematics", output: "Matematika" }
Here, the "Translation Machine" acts like a function that transforms "Mathematics" (input) into "Matematika" (output).
Now, what if we want to do the reverse? Translate "Matematika" back into "Mathematics"?
This reverse process is the basis of the **inverse function** concept.
## Definition of Inverse Function
An inverse function is a function that "reverses" the operation of an initial function. If function $$f$$ maps element $$x$$ from domain $$A$$ to element $$y$$ in codomain $$B$$, then its inverse function, denoted as $$f^{-1}$$ (read "f inverse"), maps element $$y$$ from $$B$$ back to element $$x$$ in $$A$$.
Visible text: An inverse function is a function that "reverses" the operation of an initial function. If function maps element from domain to element in codomain , then its inverse function, denoted as (read "f inverse"), maps element from back to element in .
Mathematically:
```math
y = f(x) \iff x = f^{-1}(y)
```
In other words, if $$f$$ changes $$x$$ to $$y$$, then $$f^{-1}$$ changes $$y$$ back to $$x$$. The inverse function "undoes" the effect of the original function.
Visible text: In other words, if changes to , then changes back to . The inverse function "undoes" the effect of the original function.
**Important:** The notation $$f^{-1}(x)$$ does **not** mean $$\frac{1}{f(x)}$$ (multiplicative inverse or reciprocal). It is special notation for the inverse function.
Visible text: **Important:** The notation does **not** mean (multiplicative inverse or reciprocal). It is special notation for the inverse function.
## Condition for a Function to Have an Inverse
Not all functions have an inverse function. For a function $$f$$ to have an inverse function $$f^{-1}$$, the function $$f$$ must be **bijective**. A bijective function is both **injective** (one-to-one) and **surjective** (onto).
Visible text: Not all functions have an inverse function. For a function to have an inverse function , the function must be **bijective**. A bijective function is both **injective** (one-to-one) and **surjective** (onto).
- **Injective (One-to-one):** Every distinct element in the domain maps to a distinct element in the codomain. No two different inputs produce the same output.
- **Surjective (Onto):** Every element in the codomain is the result of mapping from at least one element in the domain. All possible outputs occur.
If function $$f$$ is not bijective, its inverse relation might exist, but that relation will not be a function.
Visible text: If function is not bijective, its inverse relation might exist, but that relation will not be a function.
## Determining the Formula for an Inverse Function
To find the formula for the inverse function $$f^{-1}(x)$$ from a function $$f(x)$$, you can follow these steps:
Visible text: To find the formula for the inverse function from a function , you can follow these steps:
1. Replace $$f(x)$$ with $$y$$.
2. Swap the positions of the variables $$x$$ and $$y$$ in the equation.
3. Solve the equation for $$y$$ in terms of $$x$$.
4. Replace $$y$$ with $$f^{-1}(x)$$ to get the inverse function formula.
Visible text: 1. Replace with .
2. Swap the positions of the variables and in the equation.
3. Solve the equation for in terms of .
4. Replace with to get the inverse function formula.
**Example:**
Find the inverse function of $$f(x) = 2x + 3$$.
Visible text: Find the inverse function of .
1. Replace $$f(x)$$ with $$y$$:
```math
y = 2x + 3
```
2. Swap $$x$$ and $$y$$:
```math
x = 2y + 3
```
3. Solve for $$y$$:
```math
x - 3 = 2y
```
```math
\frac{x - 3}{2} = y
```
4. Replace $$y$$ with $$f^{-1}(x)$$:
```math
f^{-1}(x) = \frac{x - 3}{2}
```
Visible text: 1. Replace with :
2. Swap and :
3. Solve for :
4. Replace with :
So, the inverse function of $$f(x) = 2x + 3$$ is $$f^{-1}(x) = \frac{x - 3}{2}$$.
Visible text: So, the inverse function of is .
## Graph of a Function and Its Inverse
The graph of the inverse function $$f^{-1}(x)$$ is a reflection of the graph of the original function $$f(x)$$ across the line $$y = x$$.
Visible text: The graph of the inverse function is a reflection of the graph of the original function across the line .
For example, let's look at the graph of $$f(x) = 2x + 3$$, its inverse $$f^{-1}(x) = \frac{x - 3}{2}$$, and the identity line $$y=x$$.
Visible text: For example, let's look at the graph of , its inverse , and the identity line .
Component: LineEquation
Props:
- title: Graph of $$f(x)$$ and its Inverse
Visible text: Graph of and its Inverse
- description: The graph shows the function $$f(x)$$, its inverse{" "}
$$f^{-1}(x)$$, and the line $$y=x$$ as
the line of reflection.
Visible text: The graph shows the function , its inverse{" "}
, and the line as
the line of reflection.
- data: [
{
// f(x) = 2x + 3
points: Array.from({ length: 11 }, (_, i) => {
const x = i - 5;
return { x, y: 2 * x + 3, z: 0 };
}),
color: getColor("ORANGE"),
labels: [
{
text: "f(x) = 2x + 3",
at: 5,
offset: [-2.5, 0, 0],
},
],
},
{
// f⁻¹(x) = (x - 3) / 2
points: Array.from({ length: 11 }, (_, i) => {
const x = i - 5;
return { x, y: (x - 3) / 2, z: 0 };
}),
color: getColor("EMERALD"),
labels: [
{
text: "f^{-1}(x) = (x-3)/2",
at: 10,
offset: [0.5, 0.5, 0],
},
],
},
{
// y = x (Identity Line)
points: Array.from({ length: 11 }, (_, i) => {
const x = i - 5;
return { x, y: x, z: 0 };
}),
color: getColor("PINK"),
labels: [
{
text: "y = x",
at: 10,
offset: [0.5, 0.5, 0],
},
],
},
]