One to One Function — Definition, Graph, Examples
One to One Function — Definition, Graph, Examples
What Is a One to One Function?
A one to one function is a function that sends distinct inputs to distinct outputs: if x1≠x2, then f(x1)≠f(x2). Equivalently — whenever two outputs are equal, the inputs must have been equal too: f(x1)=f(x2)⟹x1=x2. The formal name is an injective function, or an injection.
How Do You Know if a Function Is One to One?
There are two standard tests — one visual, one algebraic — and they agree with each other.
The horizontal line test (visual). Look at the graph. If no horizontal line crosses it more than once, the function is one to one.
The algebraic method (proof). Set f(x1)=f(x2) and solve. If the algebra forces x1=x2, the function is one to one.
What Does a One to One Function Look Like? (The Graph)
A one to one graph never doubles back to the same height. Straight lines with non-zero slope qualify; so do cubics like f(x)=x3 and exponentials like f(x)=ex, because they climb (or fall) without ever flattening and returning.
Why Only One to One Functions Have an Inverse
A function has an inverse if and only if it is one to one. This is why f(x)=x2 has no inverse over all real numbers, but does once you restrict it to x≥0: restricting the domain throws away the duplicate arm and leaves a one to one piece.
Examples of One to One Function
Example 1
Is f(x)=5x−7 one to one?
Set the outputs equal:
5x1−7=5x2−7.
Add 7 to both sides, then divide by 5: x1=x2.
Final answer: yes, f(x)=5x−7 is one to one.
Example 2
Is f(x)=x2 one to one over all real numbers?
Final answer: no, f(x)=x2 is not one to one on R.
Example 3
Is f(x)=x3 one to one?
Final answer: yes. The graph of x3 never flattens and doubles back, so it also passes the horizontal line test.
Example 4
Is f(x)=1/(x+2) one to one?
Set the outputs equal:
1/(x1+2)=1/(x2+2).
Cross-multiplying gives x2+2=x1+2, so x1=x2.
Final answer: yes, this rational function is one to one on its domain.
Example 5
Find the inverse of the one to one function f(x)=3x−4.
Write y=3x−4, swap the variables, then solve for y: y=(x+4)/3.
Final answer: f−1(x)=(x+4)/3.
Example 6
Is a simple cipher one to one?
Final answer: yes — and being one to one is exactly what makes the message recoverable.
Where One to One Functions Earn Their Keep
- Cryptography and encoding. Any reversible code must be one to one — the shift cipher is a simple version of the bijections behind serious encryption.
- Databases and identifiers. A primary key works because the map from record to ID is one to one.
- Inverse functions everywhere. Logarithms exist because f(x)=ex is one to one; arcsine exists only on the restricted interval where sine is one to one.
Key Takeaways
- A one to one function sends distinct inputs to distinct outputs.
- The horizontal line test confirms it visually; setting f(x1)=f(x2) and solving confirms it algebraically.
- A function has an inverse if and only if it is one to one.
- The most common error is confusing the horizontal line test (one to one) with the vertical line test (is a function).
- Injectivity depends on the domain — the same formula can be one to one on one interval and not on another.