What is the difference between one-to-one and onto?

One-to-one: Each input maps to a unique output., Onto: Every element in the codomain has a pre-image.

Is every linear function one-to-one?

Yes, unless the slope is zero (constant function).

Is the function f(x) = ln x one-to-one?

Yes, it is strictly increasing and passes the horizontal line test.

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One to One function

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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 $x_{1}, x_{2} \in A$ ,
$f (x_{1}) = f (x_{2}) \Rightarrow x_{1} = x_{2}$ .

Alternatively, if $x_{1} \neq = x_{2}, t h e n f (x_{1}) \neq = f (x_{2})$ .

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 $f (x_{1}) = f (x_{2})$

$\Rightarrow 3 x_{1} + 1 = 3 x_{2} + 1$

Subtract 1 from both sides:

$\Rightarrow 3 x_{1} = 3 x_{2}$

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

$\Rightarrow f (3) = ∣3∣ = 3, \Rightarrow f (- 3) = ∣ - 3∣ = 3$

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

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

Example 5: Check whether $f (x) = x^{3}$ 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 $f (x) = x$ 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 $f (x) = sin x f or x \in [0, 2 π]$ . 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.

Join ALLEN!

(Session 2026 - 27)

Name

Mobile Number

Class

Choose class

Goal

Choose your goal

Preferred Programs

Preferred Mode

State

Choose State

I agree to

I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address.

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 $x_{1}, x_{2} \in A$ ,
$f (x_{1}) = f (x_{2}) \Rightarrow x_{1} = x_{2}$ .

Alternatively, if $x_{1} \neq = x_{2}, t h e n f (x_{1}) \neq = f (x_{2})$ .

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 $f (x_{1}) = f (x_{2})$

$\Rightarrow 3 x_{1} + 1 = 3 x_{2} + 1$

Subtract 1 from both sides:

$\Rightarrow 3 x_{1} = 3 x_{2}$

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

$\Rightarrow f (3) = ∣3∣ = 3, \Rightarrow f (- 3) = ∣ - 3∣ = 3$

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

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

Example 5: Check whether $f (x) = x^{3}$ 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 $f (x) = x$ 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 $f (x) = sin x f or x \in [0, 2 π]$ . 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.

Frequently Asked Questions

What is the difference between one-to-one and onto?

Is every linear function one-to-one?

Is the function f(x) = ln x one-to-one?

Frequently Asked Questions

What is the difference between one-to-one and onto?

Is every linear function one-to-one?

Is the function f(x) = ln x one-to-one?

Join ALLEN!

One-to-One Function

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

6.0Example of a One-to-One Function in Real Life

Table of Contents

Join ALLEN!

One-to-One Function

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

6.0Example of a One-to-One Function in Real Life

Table of Contents