## Definition of a Function

Let's look into the definition of a function.

A function is a type of mathematical relationship where an input creates an output.

Let's consider a couple of examples.

Some examples of types of functions include:

- \(f(x)=x^2\)
- \(g(x)= x^4+3\)

## Algebraic functions

Algebraic functions involved the variables and constants connected through different operations such as addition, subtraction, multiplication, division, exponentiation, etc. Let's learn about the algebraic function with its definition, types, and examples.

An algebraic function is a type of function that contains algebraic operations.

Some examples of these functions.

- \(f(x)=2x+5\)
- \(f(x)=x^3\)
- \(f(x)=2x^2+x-2\)

Algebraic functions can be plotted on a graph, each type of function creates a different type of graph.

## Different types of function graphs

The different types of functions can create different types of graphs, each with its characteristics.

### Even functions

A function is said to be even when \(f(-x)=f(x)\). An even function creates a graph where the graph line is symmetrical about the y-axis.

Some examples of even functions include, \(x^2, x^4\) and \(x^6\).

Some different types of functions can also be even, such as trigonometric functions. An example of an even trigonometric function is \(\cos(x)\).

\(\cos(-x)=\cos(x)\)

### Odd functions

A function is said to be odd when \(f(-x)=-f(x)\). An odd function creates a graph where the graph line is symmetrical about the origin.

Some examples of odd functions include, \(x\), \(x^3\) and \(x^5\).

Just like even functions, other functions can be odd, like the \(sin(x)\) function.

\(\sin(-x)=-\sin(x)\)

## Quadratic function

The word ''quad'' in the quadratic functions means ''a square''. In short, they are square functions. They are used in various fields of science and engineering. When plotted on a graph, they obtain a parabolic shape. Let's look into the definition of quadratic functions with examples.

A quadratic function is a type of function that is written in the form:

\[f(x)=ax^2+bx+c\]

You can identify a function to be quadratic if its highest exponent is 2.

Some examples of quadratic equations include:

- \(f(x)=2x^2+2x-5\)
- \(f(x)=x^2+4x+8\)
- \(f(x)=6x^2+5x-3\)

To find out more about these functions, see Forms of Quadratic functions.

## Injective, surjective, and bijective functions

Since a function is a relation between a domain and range, injective, surjective, and Bijective Functions are differentiated by that relation. To demonstrate this we can look at mappings, this will show us the different relationships each type of function has with the domain and range.

### Injective Functions

An injective function has many properties;

Only one element from the domain will point to one element in the range.

There may be elements in the range that do not have a pair in the domain.

This type of mapping is also known as 'one to one'.

To find out more visit, Injective Functions.

### Surjective Functions

A surjective function has many properties;

- All elements in the domain will have a match in the range.
- There may be an element in the range that matches with more than one of the elements in the domain.
- There won't be any elements in the range that have no match.

To find out more visit, Surjective functions.

### Bijective Functions

A bijective function has many properties;

It is a combination of injective and Surjective functions.

There is a perfect amount of elements in both the domain and range that match, there are no elements that are left out.

To find out more visit, Bijective Functions.

**Input of a function: **An **input **to a function is a value that can be plugged into a function so that a valid output is generated, and the function exists at that point. These are our x-values in a function.

**Domain of a function:** The **domain **of a function is the set of all the possible inputs of a function. The domain is as much of the set of all real numbers as possible. The set of all real numbers can be written as \(\mathbb{R}\) for short.

**Output of a function:** An **output **to a function is what we get back once the function is evaluated at the input. These are our y-values in a function.

**Codomain of a function: **The **codomain **of a function is the set of all possible outputs of a function. In calculus, a function's codomain is the set of all real numbers, \(\mathbb{R}\), unless stated otherwise.

**Range of a function: **The **range **of a function is the set of all **actual **outputs of a function. The range is a subset of the codomain. We will consider range much more often than codomain.

It is important not to get codomain and range confused. The range of a function is a subset of its codomain. In practice, we will consider a function's range much more frequently than the codomain.

## Types of exponential functions

Exponential functions help you in finding bacterial growth or decay, population growth or decay, rise or fall in the prices, compounding of money, etc. Let's look into the definition of exponential functions.

An exponential function has a constant as its base and a variable as its exponent. It can be written in the form \(f(x)=a^x\), where \(a\) is a constant and \(x\) is a variable.

Let's consider an example.

Some examples of exponential functions include:

- \(f(x)=5^x\)
- \(f(x)=4^{2x}\)
- \(f(x)=\frac{1}{3}^x\)

There are two different results of exponential functions; exponential growth or exponential decay. When this function is graphed, exponential **growth** can be identified by an **increasing** graph. Exponential **decay** can be identified by a **decreasing** graph.

## Types of functions with examples

Identify the type of function: \(f(x)=x^2\).

**Solution:**

Here \[ \begin {aligned} f(x) & =x^2 \\ f(-x) & =(-x)^2 \\ f(-x) & =x^2 \\ \end {aligned} \]

Since \(f(x)=f(-x)=x^2\)

This is an **even function**.

Identify the type of function: \(f(x)=x^5\).

**Solution:**

Here \[ \begin {aligned} f(x) & =x^5 \\ f(-x) & =(-x)^5 \\ f(-x) & =-x^5 \\ \end {aligned} \]

Since \(f(x)≠ f(-x)\)

This is an **odd function**.

Identify the type of function: \(f(x)=2x^2+4x+3\).

**Solution:**

This is a quadratic function, it is written in the correct form for a **quadratic function** and its highest exponent is \(2\).

Identify the type of function: \(f(x)=8^x\).

**Solution:**

This is an **exponential function**, the base is a constant, that is \(8\) and the power is a variable, that is \(x\).

## Types of Functions - Key takeaways

- There are many different types of functions, and each different function carries different properties.
- An even function can give you a symmetrical line on a graph about the \(y-\)axis.
- When graphed, an odd function gives a symmetrical line about the origin.
- Injective, surjective and bijective functions can all be differentiated by their mapping.

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##### Frequently Asked Questions about Types of Functions

What are examples of types of mathematical functions?

Some examples of types of mathematical functions include;

- Even functions
- Odd functions
- Injective functions
- Surjective functions
- Bijective functions

What are linear functions?

A linear function is a type of function where its graph creates a straight line.

What are the basic functions?

The basic functions include, linear functions, square functions, odd functions and even functions.

What are power functions in mathematics?

In mathematics, a power function has a variable base and constant exponent.

What are the different types of functions?

The different types of functions include; even functions, odd functions, injective functions, surjective functions, and bijective functions. These functions all have different properties.

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