One-to-one: Each input maps to a unique output., Onto: Every element in the codomain has a pre-image.
Yes, unless the slope is zero (constant function).
Yes, it is strictly increasing and passes the horizontal line test.
Join ALLEN!
(Session 2026 - 27)
Choose class
Choose your goal
Preferred Mode
Choose State
One-to-One Function
What is a One-to-One Function?
A one-to-one function (also called an injective function) is a function in which each output value is connected to exactly one input value. In simpler terms, no two different inputs produce the same output.
1.0One-to-One Function Definition
The definition of one-to-one function is:
A function f: A \rightarrow B is said to be one-to-one if for every x1,x2∈A, f(x1)=f(x2)⇒x1=x2.
Alternatively, if x1=x2,thenf(x1)=f(x2).
2.0How to Identify a One-to-One Function?
You can check whether a function is one-to-one by:
1. Algebraic Method:
Assume f(x1)=f(x2) and show that x1=x2.
2. Horizontal Line Test (Graphical Method):
If no horizontal line intersects the graph of the function at more than one point, then it is a one-to-one function graph.
3.0One-to-One Function Graph
Let’s take the function f(x) = 2x + 3.
It’s a straight line.
Every x-value maps to a unique y-value.
Passes the horizontal line test.
In this graph, no horizontal line cuts the curve more than once ⇒ It is one-to-one.
4.0Solved Examples on One-to-One Function
Example 1: Determine whether the function f(x) = 3x + 1 is a one-to-one function.
Solution: Let f(x1)=f(x2)
⇒3x1+1=3x2+1
Subtract 1 from both sides:
⇒3x1=3x2
Divide by 3:
⇒x1=x2
Since equal outputs imply equal inputs, the function is one-to-one.
Example 2: Check if the function f(x)=x2 is one-to-one.
Solution: Let’s test with values:
f(2) = 4, f(-2) = 4
But 2 ≠ −2, and yet f(2) = f(−2)
This violates the definition of one-to-one function.
Therefore, f(x)=x2 is not one-to-one over all real numbers.
Example 3: Is f(x)=2x a one-to-one function?
Solution: Let f(x1)=f(x2)
⇒2x1=2x2
Since the base is the same and positive (≠1),
⇒x1=x2
So, this function satisfies the one-to-one condition.
Therefore, f(x)=2x is a one-to-one function.
Example 4: Is f(x) = |x| a one-to-one function?
Solution: Let’s test with x = 3 and x = –3
⇒f(3)=∣3∣=3,⇒f(−3)=∣−3∣=3
But 3 ≠ −3, and still the outputs are equal.
Hence, the function is not one-to-one.
Example 5: Check whether f(x)=x3 is one-to-one.
Solution: Let f(x1)=f(x2)
⇒x13=x23
Take cube root on both sides:
⇒x1=x2
The function satisfies the condition.
Therefore, f(x)=x3 is a one-to-one function.
5.0Algebraic Practice Questions on One-to-One Functions
Q1. Determine whether the function f(x) = 3x - 7 is one-to-one.
Q2. Is the function f(x)=x2+2 one-to-one? Justify your answer.
Q3. Let f(x)=x1. Is this function one-to-one on its domain?
Q4. Is f(x) = |x| a one-to-one function? Explain why or why not.
Q5. Prove whether the function f(x)=x is one-to-one for x ≥ 0.
Graphical Practice Questions
Q6. Consider the graph of f(x)=x3. Does it pass the horizontal line test?
Q7. Draw the graph of f(x) = 2x + 5. Is it one-to-one?
Q8. Sketch f(x)=sinxforx∈[0,2π]. Is this function one-to-one over that interval?
Q9. Use a horizontal line test to determine if the function f(x)=x4 is one-to-one.
6.0Example of a One-to-One Function in Real Life
Student ID System: Every student has a unique ID number.
Each ID → one student
No two students share the same ID ⇒ One-to-one function.
Table of Contents
1.0One-to-One Function Definition
2.0How to Identify a One-to-One Function?
3.0One-to-One Function Graph
4.0Solved Examples on One-to-One Function
5.0Algebraic Practice Questions on One-to-One Functions