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# Inverse Functions — Definition, Steps, Examples

TL;DR  
An inverse function, written f−1, undoes the original function — feed an output back in and you recover the original input. This article gives the definition, the four-step method to find an inverse, the verification by composition, the reflection-over-y=x graph, why only bijective functions are invertible, and six worked examples.

## What Is An Inverse Function?  
The **inverse function** of f, written f−1, is the function that **reverses** f: if f sends x to y, then f−1 sends y back to x. The defining relationship is the pair of compositions:  
f−1(f(x)) = x and f(f−1(x)) = x  
In words: do f then f−1 (or the reverse), and you are back where you started. Applying a function and then its inverse is the [identity function](/content/math/algebra/identity-function/index.html) — the do-nothing map.  
The inverse also swaps the roles of domain and range: the **domain of f becomes the range of f−1**, and the **range of f becomes the domain of f−1.** This is the function-level version of the [inverse relation](/content/math/algebra/inverse-relation/index.html), which reverses every ordered pair.

## Which Functions Have An Inverse?  
A high-frequency search is _does every function have an inverse?_ The answer is no, and the reason is precise: **a function has an inverse if and only if it is bijective** — both one-one and onto.  
- **It must be one-one (injective).** If two inputs share an output, the reverse map can't decide which input to return. See the [one-to-one function](/content/math/algebra/one-to-one-function/index.html).  
- **It must be onto (surjective)** its codomain, so every value you might feed to f−1 actually came from somewhere.  
Together, those are the conditions for a [bijective function](/content/math/algebra/bijective-function/index.html). When a function isn't one-one — like f(x)=x² over all reals — we often **restrict the domain** (to x≥0) so that the restricted piece is bijective and an inverse exists.

## How Do You Find The Inverse Of A Function?  
The standard method has four steps. _How do you find the inverse of a function algebraically?_ is one of the most-searched phrasings, so here it is in full:  
1. **Replace** f(x) with y.  
2. **Swap** x and y everywhere.  
3. **Solve** the new equation for y.  
4. **Rename** y as f−1(x).

The swap in step 2 is the heart of it — it encodes "outputs become inputs." Each worked example below runs these four steps.

## What Does The Graph Of An Inverse Function Look Like?  
The graph of f−1 is the **reflection of the graph of f across the line y=x.** Because the inverse swaps each point (a,b) into (b,a), the whole curve flips over that diagonal. This gives a fast visual check: if you fold the paper along y=x and the two graphs land on each other, they are inverses. It also explains why a function passes the horizontal line test exactly when its inverse passes the vertical line test — reflection turns one test into the other.

## Examples Of Inverse Functions  
The six examples build from a clean line to a rational function and a restricted-domain case, with one deliberate wrong turn.

### Example 1  
**Find the inverse of f(x)=2x+3.**  
Replace f(x) with y: y=2x+3.  
Swap x and y: x=2y+3.  
Solve for y: x−3=2y, so y=(x−3)/2.  
Rename: f−1(x)=(x−3)/2.

### Example 2  
**Find the inverse of f(x)=x³−1, then verify it.**  
Run the four steps instead.  
1. y=x³−1.  
2. Swap: x=y³−1.  
3. Solve: y³=x+1, so y=³√(x+1).  
4. Rename: f−1(x)=³√(x+1).  
Verify: f(f−1(x))=x. Correct.

### Example 3  
**Find the inverse of f(x)=4x+1/3x−2.**  
1. y=4x+1/3x−2.  
2. Swap: x=4y+1/3y−2.  
3. Clear the denominator: x(3y−2)=4y+1.  
4. Solve: y=(2x+1)/(3x−4).  
5. Rename: f−1(x)=(2x+1)/(3x−4).

### Example 4  
**Verify that f(x)=5x−7 and g(x)=x+7/5 are inverses.**  
1. Compose one way: f(g(x))=x.  
2. Compose the other way: g(f(x))=x.  
Both compositions return x, so f and g are **inverses**.

### Example 5  
**Find the inverse of f(x)=x² on the restricted domain x≥0.**  
1. y=x².  
2. Swap: x=y².  
3. Solve, keeping y≥0: y=√x.  
4. Rename: f−1(x)=√x.

### Example 6  
**A temperature converter sends Celsius to Fahrenheit by F(c)=9/5c+32. Find the inverse that converts back.**  
1. y=9/5c+32.  
2. Swap: c=9/5f+32.  
3. Solve for f: f=(5/9)(c−32).  
4. The inverse is F−1(c)=(5/9)(c−32).

## Why Inverses Matter: "An Inverse Is The Mathematics Of Undoing"  
Inverse functions exist because almost every useful process needs to be reversible — and mathematics needed a precise object for "go back."  
- **They solve equations.** Solving f(x)=c is the act of applying f−1 to both sides. Logarithms invert exponentials, roots invert powers, [inverse trigonometric functions](https://en.wikipedia.org/wiki/Inverse_trigonometric_functions) invert sine and cosine.  
- **They reverse transformations.** A coordinate change, a currency conversion, or an encryption step is only useful if it can be undone — and the undo is an inverse function.  
- **They define new operations.** The logarithm was _defined_ by [John Napier](https://mathshistory.st-andrews.ac.uk/Biographies/Napier/) in 1614 as the inverse of exponential growth.

## Where Inverses Trip Students Up  
Three mistakes account for most wrong inverse answers.  
### Mistake 1: Reading f−1 as a reciprocal  
**Where it slips in:** the instant the −1 notation appears.  
**Don't do this:** write f−1(x)=1/f(x).  
**The correct way:** f−1 is the reverse map, found by swapping x and y and solving — not by flipping a fraction.

### Mistake 2: Forgetting to check the function is one-one first  
**Where it slips in:** parabolas or any many-to-one function.  
**Don't do this:** report ±√x as a function.  
**The correct way:** Restrict the domain first.

### Mistake 3: Swapping in the wrong place  
**Where it slips in:** rational and multi-step functions.  
**Don't do this:** swap x and y on one side but not the other.  
**The correct way:** replace _every_ x with y and _every_ y with x in one clean move, then solve.

## Conclusion  
- An **inverse function** f−1 undoes f: f−1(f(x))=x.
- Find it by swapping x and y and solving — never by taking a reciprocal.
- Only bijective functions are invertible; non-bijective ones need a restricted domain.
- Inverses are how we solve equations and reverse transformations.