Features: definition, types, scope of domain and video lesson (2023)

Functions are relations where each input has a specific output. This lesson covers the concepts of functions in mathematics and the different types of functions using various examples for better understanding.

Feature Related Content

• functions
• Borders, continuity and differentiability
• differentiation
• derivative applications

JEE Main 2021 Maths LIVE Paper Solutions 24. Feb Turn-1 Based on memory

What are functions in mathematics?

A function is a relation between a set of inputs and a set of allowed outputs with the property that each input is associated with exactly one output. Let A and B be any two non-empty sets; The mapping from A to B is a function only if every element in set A has an edge, only one image in set B.

Example:

Another definition of a function is that it is a relation "f" in which each element of the set "A" is associated with a single element belonging to the set "B". Also in a function there cannot be two pairs with the same first element.

A condition for a function:

DefineAand adjustBmust not be empty.

In a function, a specific input is given to get a specific output. So a functionf: A->Bmeans that f is a function ofAProB, Where fromAis a domain andBis a codomain.

• per articleawho listenedA, a∈ A,a single articleb, b∈Bexists such that (a,b)∈F.

the unique articlebwhichFRegardsa, is denoted byFa)and it's calledf of one, or the value off in one, othe image of a bass f.

• oreachVonF(Picture ofasollozo f)
• is the set of all values ​​off(x)taken together.
• area f= {j∈AND | y = f(x), for some x in X}

has a real valued functionPAGor one of its subsets as its domain. Even if your domain is tooPAGor a subset of itPAG, diceactual function.

Vertical line test:

The vertical line test is used to determine whether or not a curve is a function. If a curve intersects a vertical line at more than one point, the curve is not a function.

function representation

Functions are usually represented asf(x).

you are f(x) = x³.

We say that f of x is equal to x cubes.

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

Steps to solve functions

Question:Find the output of the function g(t) = 6t²+ 5 Em

(yo) t = 0

(ii) t = 2

Solution:

The given function is g(t) = 6t²+5

(i) At t = 0, g(0) = 6(0)²+ 5 = 5

(ii) Em t = 2, g(2) = 6(2)²+ 5 = 29

types of functions

There are different types of functions in mathematics, which are explained in more detail below. The different types of functions covered here are:

• One - a function (injective function)
• many - one role
• in function (surjective function)
• function
• polynomial function
• lineal funtion
• identical function
• quadratic function
• rational function
• algebraic functions
• cubic function
• module function
• signal function
• largest integer function
• fraction function
• even and odd function
• periodic function
• compound function
• constant function
• identity function

Elaborate:Find the missing equations in the graphs above.

Features - Video Lessons

One - a function (injective function)

If each element in the scope of a function has a specific image in the scope, the function is calledone - one function.

For exampleF; R R given by f(x) = 3x + 5 is one – one.

many - one role

On the other hand, if there are at least two elements in the domain whose images are equal, the function is said to be a many-to-one function.

For examplef : R R dada por f(x) = x²+ 1 is many one.

in function (surjective function)

A function is calledin functionif each element in the range has at least one ancestor in the domain.

function

If there is at least one element in the range that is not an image of an element in the domain, the function is Into.

(Q) Let A = {x : 1 < x < 1} = B be a function f : A B, find the nature of the given function (P). F(x) = |x|

f(x) = |1|

Solution for x = 1 & -1

Therefore, it is very important that the range of f(x) be from [-1, 1] to [0, 1], which is not equal to the codomain.

So it works.

Suppose we have a function,

\(\begin{array}{l}f(x)=\left\{\begin{matriz} x^2 & ; & x\geq 0\\ -x^2 & ; & x<0 \end{matriz }\direita.\end{matriz} \)

For different input values, we have different outputs, so it's... a function that is also handled equals its codomain, so that's it too.

polynomial function

A real valued functionf : PAG → PAGdefined by

\(\begin{matriz}{l}y = f(a) = h_{0}+h_{1}a+…..+h_{n}a^{n}\end{matriz} \)

• norte= a non-negative integer.
• odegreeof the polynomial function is the highest power in the expression.
• yes orthe degree is Null, is called a constant function.
• yes orgrade is inand is called a linear function. Example: b = a+1.
• Graph type: Always a straight line.

Therefore, a polynomial function can be expressed as:

\(\begin{matriz}{l}f(x)= a_{n}x^{n}+a_{n-1}x^{n-1}+…..+a_{1}x^{ 1}+a_{0}\end{matriz} \)

The highest power in expression is known asDegree of the polynomial function. The different types ofpolynomial functionsby grade are:

1. The polynomial function is called a constant function if the degree is zero.
2. The polynomial function is called linear if the degree is one.
3. The polynomial function is quadratic if the degree is two.
4. The polynomial function is cubic if the degree is three.

lineal funtion

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

Example: f(x) = 2x + 1 at x = 1

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

f(1) = 3

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

identical function

Of theThe functions f and g must be identicalwhat if

(a)The domain of f = domain of g

(b)The area of ​​f = the area of ​​g

(C)f(x) = g(x)∀ x ∈ D_F&D_gram

For examplef(x) = x

\(\begin{matrix}{l}g(x) = \frac{1}{1/x}\end{matrix} \)

Solution: f(x) = x is defined for all x

but

\(\begin{matrix}{l}g(x) = \frac{1}{1/x}\end{matrix} \)

So it is identical for x∈ R-{0}

quadratic function

All functions in the form y = ax²+ bx + conde a, b, c∈R, a ≠ 0 will be known as a quadratic function. The graph will be parabolic.

\(\begin{matriz}{l}\text{At}\ x=\frac{-b \pm \sqrt{D}}{2}\end{matriz} \)

in a nutshell,

A quadratic polynomial function is a polynomial of the second degree and can be expressed as;

F(x) = ax²+ bx + c, and a is nonzero.

where a, b, c are constants and x is a variable.

Example, f(x) = 2x²+ x - 1 y x = 2

Se x = 2, f(2) = 2,2²+ 2 – 1 = 9

For example: y = x²

For more information, see:quadratic function formula

rational function

These are real functions of type

\(\begin{matriz}{l}\frac{f(a)}{g(a)}\end{matriz} \)

• For examplef : PAG – {– 6} → PAGdefined by

  \(\begin{array}{l}f(a) = \frac{f(a+1)}{g(a+2)}\forall a\in P – {-6},\end{array} \)

  Is arational function.
• Graph type: asymptotes (the curves touch the lines of the axes).

algebraic functions

An algebraic equation is known as a function consisting of a finite number of terms, involving powers and roots of the independent variable x and basic operations such as addition, subtraction, multiplication, and division.

For example,

\(\begin{matriz}{l}f(x)=5x^{3}-2x^{2}+3x+6\end{matriz} \)

cubic function

A cubic polynomial function is a polynomial of the third degree and can be expressed as;

F(x) = ax³+ bx²+ cx + d y a is nonzero.

In other words, any function of the form f(x) = ax³+ bx²+ cx + d, onde a, b, c, d∈R& and ≠ 0

For example:y = x³

Domain∈R

Reach∈R

module function

the true functionf : PAG → PAGdefined by f(a) = |a| = that when that≥ 0. y f(a) = -a si a < 0 ∀ a ∈p is calledmodule function.

• DomainVonF=PAG
• reachF=P⁺tu {0}

\(\begin{matriz}{l}y=|x|=\left\{\begin{matriz} x & x\geq 0\\ -x & x<0 \end{matriz}\right.\end{ Vielfalt} \)

Domain: R

Range: [0,∞)

signal function

the true functionf : PAG → PAGis defined by

\(\begin{matriz}{l}\left\{\begin{matriz}\frac{\left | f(a) \right |}{f(a)}, a\neq 0 \\ 0, a= 0 \end{matriz}\right.= \left\{\begin{matriz} 1, cae x>0\\ 0, cae x=0\\ -1, cae x<0\end{matriz}\right. \end{matriz} \)

dicesignal functionosignal function. (indicates the sign of the real number)

• Domain of definition of f =PAG
• range f = {1, 0, - 1}

Example: Character(100) = 1

sign (register 1) = 0

character(x)²1) =1

largest integer function

the true functionf : PAG → PAGdefined byf(a) = [a], a ∈ PAGtakes the value of the largest integer that is less than or equal toa,dicelargest integer function.

• Then f(a) = [a] = – 1 for – 1 ⩽ a < 0
• f(a) = [a] = 0 for 0 ⩽ a < 1
• [a] = 1 for 1 ⩽ a < 2
• [a] = 2 for 2 ⩽ a < 3 and so on...

olargest integer functionalways gives full power. The largest integral value obtained from the input is the output.

For example: [4,5] = 4

Domain∈R

Reach∈all

fraction function

{x} = x – [x]

It always gives a fractional value as output.

For example:- {4,5} = 4,5 – [4,5]

= 4,5 – 4 = 0,5

{6,99} = 6,99 – [6,99]

= 6,99 – 6 = 0,99

{7} = 7 – [7] = 7 –7 = 0

even and odd function

If f(x) = f(-x) then the function is even & f(x) = -f(-x) then the function is odd

Example 1:

f(x) = x²sin

f(-x) = -x²sin

Here, f(x) = -f(-x)

It is a strange feature.

Example 2:

\(\begin{matriz}{l}f(x)={{x}^{2}}\end{matriz} \)

mi

\(\begin{matriz}{l}f(-x)={{x}^{2}}\end{matriz} \)

f(x) = f(-x)

It is an even function.

periodic function

A function is said to be periodic if there exists a positive real number T such that f(u – t) = f(x) for every region x ε.

For example f(x) = sin x

f(x + 2π) = sen (x + 2π) = sen x fundamental

then the period of sin x is 2π

compound function

Let A, B, C be three non-empty sets.

you are f:a→B y G: G→C can be two functions, so gof : A→C. This function is called data composition of f and g gof(x) = g(f(x)).

For example f(x) = x²&g(x) = 2x

f(g(x)) = f(2x) = (2x)²= 4x²

g(f(x)) = g(x²) = 2x²

constant function

The functionf : PAG → PAGis defined bysecond = f(x) = D,a ∈PAG, Where fromDis a constant∈ PAG, is a constant function.

• domain off = pag
• reachf = {D}
• Graph Type: A straight line parallel to the x-axis.

In simple terms, the polynomial of degree 0, where f(x) = f(0) = a₀= do. Regardless of the input, the output is always a constant value. The graph for this is a horizontal line.

identity function

PAG= my real numbers

The functionf : PAG → PAGdefined byb = f(a) = unfor each∈ PAGis called the identity function.

• domain ofF=PAG
• reachF=PAG
• Graph Type: A straight line through the origin.

video capabilities