One-to-One Functions

Definition

A one-to-one function (also called an injective function) is a function where each output value corresponds to only one unique input value. In other words, no two different input values produce the same output. Mathematically, a function 𝑓(𝑥) is one-to-one if: $$ 𝑓(𝑎)=𝑓(𝑏)⇒𝑎=𝑏 $$ for all values of 𝑎 and 𝑏 in the domain of 𝑓.

Graphical Interpretation

A function is one-to-one if it passes the **horizontal line test**:

✅ Passes: If every horizontal line intersects the graph at most once.
❌ Fails: If any horizontal line intersects the graph more than once.

Examples of One-to-One Functions

1. Linear Function

Let \[ f(x) = 2x + 3 \]

Assume: \[ f(a) = f(b) \]:

\[ 2a + 3 = 2b + 3\] \[ \Rightarrow a = b \]

Since, $ a = b $, this function is one-to-one.

2. Exponential Function

Let \[ f(x) = 3^x \]

Assume: \[ f(a) = f(b) \]:

\[ 3^a = 3^b \] \[ \Rightarrow a = b \]

Since the exponential function is strictly increasing, this implies 𝑎=𝑏.
Hence, $𝑓(𝑥)=3^𝑥 $ is one-to-one.

3. Cubic Function

Let \[ f(x) = x^3\]

Assume \[ f(a) = f(b) \]:

\[ a^3 = b^3 \] \[ \Rightarrow a = b \qquad (Taking \quad cube \quad root \quad both \quad sides) \]

Since $ a = b $, this function is one-to-one.

Graphical Check: The graph of $𝑦=𝑥^3$ passes the horizontal line test—every horizontal line cuts the graph only once. ✅

4. Logarithmic Function

Let \[ f(x) = log_2 (x)\]

Assume \[ f(a) = f(b) \]:

\[ log_2(a) = log_2(b) \] \[ \Rightarrow a = b \qquad (By \quad the \quad properties \quad of \quad logarithms) \]

Since $ a = b $, this function is one-to-one.

Graphical Check: The graph of $𝑦= log_2(x)$ passes the horizontal line test. Since The logarithmic function increases throughout its domain and passes the horizontal line test. ✅

Functions That Are NOT One-to-One

1. Quadratic Function

Let \[ f(x) = x^2 \]

Inputs:\[ f(2) = 4 \] and \[ f(-2) = 4 \]

→ Two inputs give the same output.

Since, different inputs give the same output, therefore, $𝑓(𝑥)=𝑥^2$ is not one-to-one.

This function **fails** the horizontal line test.

2. Absolute Value Function

Let \[ f(x) = | x | \]

Inputs: \[ f(3) = 3 \] and \[ f(-3) = 3 \]

→ Two inputs give the same output.

Since, different inputs give the same output, therefore, $𝑓(𝑥)= | x |$ is not one-to-one.

This function **fails** the horizontal line test.

3. Sine Function

Let \[ f(x) = \sin x \]

Since, \[ \sin(\pi/6) = 1/2 \] and \[ \sin(5\pi/6) = 1/2 ,\]

→ Two inputs give the same output.

So the function $ f(x) = \sin x $ is **not** one-to-one.

How to Determine if a Function is One-to-One

Algebraic Method:
Assume $ 𝑓(𝑎)=𝑓(𝑏) $ and solve, then check if it leads to $𝑎=𝑏 $. If it is $ a = b $, then it is one-to-one; otherwise, it is not.
Graphical Method: Use the horizontal line test
—if any horizontal line cuts the graph more than once, the function is not one-to-one.

✅ If a horizontal line touches the graph only once everywhere, the function is one-to-one.
❌ If a horizontal line touches the graph at multiple points, the function is NOT one-to-one.

Conclusion

A one-to-one function ensures that each input has a unique output. This property is important in inverse functions since only one-to-one functions have well-defined inverses.

Interactive Example

Click "Next Problem" to generate a random function and check whether it is **one-to-one** or **not**.