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# Types of Functions

Have you ever considered how you throw a ball? The way in which it falls can be modeled by a quadratic function. Maybe you've wondered how the population may change over time. Well, that can be calculated using exponential functions. There are many different types of functions that are seen in everyday life! In this article, you will be learning about different types of functions.

## 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.

Fig. 1. Even function graph.

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.

Fig. 2. Odd function graph.

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.

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

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.

Fig. 3. Injective, Surjective, and Bijective Mappings.

### 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.

Some examples of types of mathematical functions include;

• Even functions
• Odd functions
• Injective functions
• Surjective functions
• Bijective functions

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

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

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

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

## Final Types of Functions Quiz

Question

What is an odd function?

A function, f(x) is an odd function if

f(-x) = -f(x), for all x, x∈R

Show question

Question

What does an odd function iook iike on a graph?

An odd function is symmetric about the origin, when plotted on a graph

Show question

Question

What is an even function?

A function, f(x) is an even function if

f(x) = f(-x), for all x, x∈R

Show question

Question

What does an odd function look like on a graph?

An odd function is not symmetric about the y-axis.

Show question

Question

State whether the following statement is true or false:

The sum of two even functions is an even function.

True

Show question

Question

State whether the following statement is true or false:

The sum of two odd functions is an even function.

False

Show question

Question

State whether the following statement is true or false:

The product of two odd functions is an odd function.

False

Show question

Question

State whether the following statement is true or false:

Odd functions are symmetric with respect to the origin.

True

Show question

Question

State whether the following statement is true or false:

The sum of two even functions can be either odd or even.

False

Show question

Question

State whether the following statement is true or false:

The difference between two even functions is an odd function.

False

Show question

Question

State whether the following statement is true or false:

The product of two even functions is always an even function.

True

Show question

Question

State whether the following statement is true or false:

All trigonometric functions are odd functions.

False

Show question

Question

Is the following function an even function:

f(x)=sin(x)

No

Show question

Question

Is the following function an even function:

f(x)=cos(x)

Yes

Show question

Question

Is the following function an even function:

f(x)=sin(x) + 5x²

No

Show question

Question

Is the following function an even function:

f(x)=csc(x) + sec(x)

No

Show question

Question

Is the following function an even function:

f(x) = x + 5

No

Show question

Question

What is an injective function?

A function f : A ⇒ B is defined to be one-to-one or injective, if the images of distinct elements of A under f are distinct.

Show question

Question

What is a surjective function?

A function f : A -->B is surjective if and only if for every element, y in B, there is at least one element, x in A such that f(x) = y.

Show question

Question

State whether the following statement is true or false :

In a surjective function, the domain and range are always identical.

False

Show question

Question

State whether the following statement is true or false :

In a surjective function, the range and co-domain are always identical.

True

Show question

Question

State whether the following statement is true or false :

An injective function is also called an onto function.

False

Show question

Question

State whether the following statement is true or false :

An injective function is also called a one-to-one function.

True

Show question

Question

State whether the following statement is true or false :

A surjective function is also called a one-to-one function.

False

Show question

Question

State whether the following statement is true or false :

An surjective function is also called an onto function.

True

Show question

Question

Is the following function surjective?

f(x)=x, x∈R

Yes

Show question

Question

Is the following function surjective?

f(x)=x, x∈N

No

Show question

Question

Is the following function injective?

f(x)=x, x∈R

Yes

Show question

Question

Is the following function injective?

f(x)=x², x∈R

No

Show question

Question

Is the following function surjective?

f(x)=x², x∈R

No

Show question

Question

Is the following function surjective?

f(x)=cos(x), x∈R

No

Show question

Question

Is the following function injective?

f(x)=sin(x), x∈R

No

Show question

Question

Is the following function injective?

f(x)=x²+5x+3, x∈R

No

Show question

Question

Is the following function injective?

f(x)=x²-36, x∈R

No

Show question

Question

Is the following function injective?

f(x)=cosec(x) + x, x∈R

No

Show question

Question

Is the following function injective?

f(x)=5x, x∈N

Yes

Show question

Question

Is the following function surjective?

f(x)=5x, x∈N

No

Show question

Question

Is the following function injective?

f(x)=5x³, x∈N

Yes

Show question

Question

Is the following function injective?

f(x)=2x+2, x∈N

Yes

Show question

Question

Is the mapping from student to roll number a bijective function?

Yes

Show question

Question

Which of the following is true for a bijective function?

It is injective

Show question

Question

What is a bijective function?

A function f : A ⇒ B is bijective if, for every y in B, there is exactly one x in A such that f(x) = y.

Show question

Question

State whether the following statement is true or false:

In a bijective function, the domain and range are identical.

False

Show question

Question

State whether the following statement is true or false:

In a bijective function, the co-domain and range are identical.

True

Show question

Question

As per the horizontal test on bijective function, how many intersecting points with the horizontal line should occur?

Exactly one

Show question

Question

Is the following function bijective:

f(x)=2x³

Yes

Show question

Question

Is the following function bijective:

f(x)=2x³, the domain and co-domain are the set of all natural numbers

No

Show question

Question

Is the following function bijective:

f(x)=5x, the domain and co-domain are the set of all real numbers

Yes

Show question

Question

Is the following function bijective:

f(x)=x, the domain and co-domain are the set of all natural numbers

Yes

Show question

Question

What type of functions can have inverse functions?

Injective functions

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