# Types of Functions

**Introduction**

A function is a special type of relation where each input has exactly one output. Functions are essential tools in mathematics and have a wide range of applications in various fields, including physics, chemistry, engineering, and economics.

**Types of Functions**

There are various types of functions, each with distinct properties and applications. Here are some of the most common types of functions:

**1. One-to-One Functions (Injective Functions)**

A one-to-one function, also known as an injective function, is a function where each input corresponds to a unique output. In other words, no two distinct inputs have the same output.

**Properties:**

- Each input has a unique output.
- The horizontal line test can be used to identify one-to-one functions.

**Examples:**

- (for )
- (for )

**2. Onto Functions (Surjective Functions)**

An onto function, also known as a surjective function, is a function where every output has a corresponding input. In other words, every element in the range of the function has at least one input that maps to it.

**Properties:**

- Every output has at least one corresponding input.
- The range of the function is equal to the codomain of the function.

**Examples:**

- (for )
- (for )

**3. Bijective Functions**

A bijective function is a function that is both one-to-one and onto. In other words, each input has a unique output, and every output has a corresponding input. Bijective functions are also known as invertible functions.

**Properties:**

- Each input has a unique output.
- Every output has a corresponding input.
- The function has an inverse function.

**Examples:**

- (for )
- (for )

**4. Increasing and Decreasing Functions**

An increasing function is a function where the output increases as the input increases. Conversely, a decreasing function is a function where the output decreases as the input increases.

**Properties:**

- Increasing function: If , then
- Decreasing function: , then

**Examples:**

- Increasing function: (for )
- Decreasing function: (for )

**5. Even and Odd Functions**

An even function is a function where for all values of in the domain of the function. Conversely, an odd function is a function where for all values of in the domain of the function.

**Properties:**

- Even function: The graph of an even function is symmetric about the y-axis.
- Odd function: The graph of an odd function is symmetric about the origin.

**Examples:**

- Even function:
- Odd function:

**Conclusion**

Understanding the different types of functions is crucial for analyzing and solving problems in mathematics and various other fields. By recognizing the properties and characteristics of each type of function, students can effectively approach mathematical problems and apply functions in real-world applications.