Functions in Mathematics

Functions are relations where each input has a particular output. In this lesson, we will cover the concepts of functions in mathematics and the different types of functions, with various examples to help with better understanding.

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Differentiation

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##What are Functions in Mathematics? A function in mathematics is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.

A function is a relation between a set of inputs, A, and a set of permissible outputs, B, such that each input is related to exactly one output. For a mapping from A to B to be a function, it must be the case that every element in set A has one and only one image in set B.

Example:

This is an example statement.

Answer:

This is an example statement.

A function can be defined as a relation “f” where each element of set “A” is mapped to only one element belonging to set “B”, and there can’t be two pairs with the same first element.

A Necessary Condition for a Function:

Set A and Set B should not be empty.

A function f: A → B denotes that f is a function from A to B, where A is the domain and B is the codomain.

For an element, a, which belongs to A (a ∈ A), is there a unique element b, which belongs to B (b ∈ B), such that (a,b) ∈ f?

The element f(a), which relates a to b, is commonly referred to as f of a, the value of f at a, or the image of a under f.

The range of f(a)

The collective values of f(x) taken together is referred to as the set.

Range of f = {y ∈ Y | y = f(x), for some x ∈ X}

A real-valued function has P or any of its subsets as its range, and if its domain is also either P or a subset of P, it is referred to as a real function.

Vertical Line Test: A way to determine if a graph of a relation is a function is by using the vertical line test. This states that if a vertical line is drawn through the graph, it should not intersect the graph more than once.

If a vertical line intersects a curve in more than one point, then the curve is not a function. This is known as the Vertical Line Test.

Representation of Functions

f(x) is a common representation of a function.

f(x) = x^3

f(x) = x^3

Function can also be represented by g(), t(), etc.

Steps for Solving Functions

Answer: The output of the function g(t) = 6t² + 5 at is 36t² + 5.

(i) t = 0

(ii) t = 2

Given:

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Solution:

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g(t) = 6t<sup>2</sup> + 5

(i) At $t = 0$, $g(0) = 6(0)^2 + 5 = 5$

(ii) At t = 2, g(2) = 6(2)² + 5 = 29

Types of Functions

The Different Types of Functions in Mathematics

Below is a list of the various types of functions in mathematics, explained in detail:

7. Linear

8. Quadratic

9. Polynomial

10. Rational

11. Exponential

12. Logarithmic

One → One Function (Injective Function)

Many-to-One Function

Onto - Function (Surjective Function)

Into() Function

Polynomial function

Linear Function

Identical Function

Quadratic Function

Rational Function

Algebraic Functions

Cubic Function

Modulus Function

Signum Function

Greatest Integer Function

Fractional Part Function

Even and Odd Function

Periodic Function

Composite Function

Constant Function

Identity Function

![Types of Function]()

Practice: Identify the equations that are missing from the above graphs.

Functions - Video Lessons

Functions and Types of Functions

Number of Functions

Even and Odd Functions

Composite and Periodic Functions

One-to-One Function (Injective Function)

If each element in the domain of a function has a distinct image in the co-domain, the function is said to be one-to-one.

For example, the function $f: R \rightarrow R$ given by $f(x) = 3x + 5$ is one-to-one.

Many-to-One Function

If there are at least two elements in the domain whose images are the same, the function is known as many to one.

![Many-to-One Function]()

For example, the function $f: \mathbb{R} \rightarrow \mathbb{R}$ given by $f(x) = x^2 + 1$ is a many-one function.

Onto - Function (Surjective Function)

A function is called an onto function if every element in the codomain has a corresponding element in the domain.

Into Function

If there exists at least one element in the co-domain which is not an image of any element in the domain, then the function will be an Injective function.

(Q) What is the nature of the function $f(x) = |x|$ given the mapping $A = {x : 1 < x < 1} = B$? (P) The nature of the given function $f(x) = |x|$ is an absolute value function.

f(x) = |1|

Solution for x = 1 and -1

Hence, it is evident that the Range of $f(x)$ from $[-1, 1]$ is $[0, 1]$, which does not equal the Co-domain.

Therefore, it is a part of the function.

![Into - Function Example]()

Let’s say we have a function,

$$f(x)=\begin{cases} x^2 & \text{if } x \geq 0\ -x^2 & \text{if } x < 0 \end{cases}$$

For different values of Input, we have different output, making it a one-to-one function. Furthermore, it maps to its co-domain, making it an onto function.

Polynomial Function

A real-valued function f : P → P defined by (\begin{array}{l}y = f(a) = h_{0}+h_{1}a+…..+h_{n}a^{n}\end{array} ), where **n ∈ N and h0 + h1 + … + hn ∈ P, for each a ∈ P, is called a polynomial function.

N: a non-negative integer

The highest power of a Polynomial function is referred to as its ‘degree’.

If the degree is zero, it is referred to as a constant function.

If the degree is 1, it’s called a linear function. Example: b = a + 1.

Graph Type: Always a Straight Line

A polynomial function can be expressed as:

(\begin{array}{l}f(x) = a_0 + a_1x + a_2x^2 + \dots + a_nx^n\end{array})

The degree of a polynomial function is the highest power of the expression. Depending on the degree, polynomial functions can be classified into the following types: Polynomial Functions.

1. A Constant function is a polynomial function with a degree of zero.

2. A polynomial function is called Linear if its degree is one.

3. The degree of a polynomial function is two if it is Quadratic.

4. The degree of a polynomial function is three if it is Cubic.

Linear Function

All functions in the form of ax + b where a, b ∈ R & a ≠ 0 are called linear functions. The graph of such a function will be a straight line. In other words, a linear polynomial function is a first-degree polynomial where the input needs to be multiplied by m and added to c. It can be expressed by f(x) = mx + c.

f(1) = 2(1) + 1 f(1) = 3

f(1) = 2.1 + 1 = 3

f(1) = 3

Another example of a linear function is y = x + 3

Identical Function

Two functions f and g are said to be identical if they produce the same output for the same input.

(a) The domain of f is equal to the domain of g

(b) The range of f is equal to the range of g.

(c) $\forall x \in D_f \cap D_g, f(x) = g(x)$

For example:

$$f(x) = x$$

(\begin{array}{l}g(x) = x\end{array} )

f(x) = x is defined for all x.

But (g(x) = \frac{1}{1/x}) is not defined when (x = 0).

Therefore, it holds true for all x in R, excluding 0.

Quadratic Function

All functions of the form y = ax² + bx + c, where a, b, c ∈ R and a ≠ 0, will be known as a Quadratic Function. The graph of this function will be parabolic.

At (x=\frac{-b \pm \sqrt{D}}{2}), the maximum or minimum value of the equation depends on the leading coefficient and is equal to (-\frac{D}{4a}), where (D) is the discriminant.