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 $x_1,x_2\in A$,
$f\left(x_1\right)=f\left(x_2\right)\Rightarrow x_1=x_2$.

Alternatively, if $x_1\ne x_2,thenf\left(x_1\right)\ne f\left(x_2\right)$.

2.0How to Identify a One-to-One Function?

You can check whether a function is one-to-one by:

1. Algebraic Method:

Assume $f(x_1)=f(x_2)$ and show that $x_1=x_2$.

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 $f(x_1)=f(x_2)$

$\Rightarrow 3x_1+1=3x_2+1$

Subtract 1 from both sides:

$\Rightarrow 3x_1=3x_2$

Divide by 3:

$\Rightarrow x_1=x_2$

Since equal outputs imply equal inputs, the function is one-to-one.

Example 2: Check if the function $f(x)=x^2$ 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, $f(x)=x^2$ is not one-to-one over all real numbers.

Example 3: Is $f(x)=2^x$ a one-to-one function?

Solution:
Let $f(x_1)=f(x_2)$

$\Rightarrow 2^{x_1}=2^{x_2}$

Since the base is the same and positive (≠1),

$\Rightarrow x_1=x_2$

So, this function satisfies the one-to-one condition.

Therefore, $f(x)=2^x$ 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

$\begin{array}{rl}& \Rightarrow f(3)=|3|=3,\\ & \Rightarrow f(-3)=|-3|=3\end{array}$

But 3 ≠ −3, and still the outputs are equal.

Hence, the function is not one-to-one.

Example 5: Check whether $f(x)=x^3$ is one-to-one.

Solution:
Let $f(x_1)=f(x_2)$

$\Rightarrow x_1^3=x_2^3$

Take cube root on both sides:

$\Rightarrow x_1=x_2$

The function satisfies the condition.

Therefore, $f(x)=x^3$ 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 $f(x)=x^2+2$ one-to-one? Justify your answer.

Q3. Let $f(x)=\frac{1}{x}$. 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 $f(x)=\sqrt{x}$ is one-to-one for x ≥ 0.

Graphical Practice Questions

Q6. Consider the graph of $f(x)=x^3$. Does it pass the horizontal line test?

Q7. Draw the graph of f(x) = 2x + 5. Is it one-to-one?

Q8. Sketch $f(x)=\sin xforx\in [0,2\pi ]$. Is this function one-to-one over that interval?

Q9. Use a horizontal line test to determine if the function $f(x)=x^4$ 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.