This document contains a summary of key concepts in mathematics functions including: - Definitions of different types of functions such as polynomial, rational, trigonometric, exponential, and logarithmic functions. - Concepts related to functions like domain, range, and notation for functions. - Examples demonstrating how to find the domain and range of functions. - Key topics in trigonometry including trigonometric functions, trigonometric tables, addition formulas, laws of sines and cosines. - Concepts of limits, average and instantaneous rates of change, techniques for finding limits, and limits involving infinity.

1. NAME: QAMAR ABBAS JAFFARI
CASTE: PATHAN
ROLL NUMBER: 2K20/LCS/
ASSIGNMENT:
qamarabbas72jaffari@gmail.com
Mathematics
Assignment
Mathematics

2. Introduction
Functions In mathematics, a function is a relation between a set of inputs and a
set of permissible outputs. Functions have the property that each input is related
to exactly one output. ... In the case of a function with just one input variable, the
input and output of the function can be expressed as an ordered pair.
Domain and Range
Domain and Range of Rational Functions .The domain of a function f(x) is the set
of all values for which the function is defined, and the range of the function is the
set of all values that f takes. A rational function is a function of the form f(x) =p(x)
q(x), where p(x) and q(x) are polynomials and q(x)≠0 .
Definition of Function
A technical definition of a function is: a relation from a set of inputs to a set of
possible outputs where each input is related to exactly one output.
Types of functions
Constant Function:
Let „A‟ and „B‟ be any two non–empty sets, then a function „f‟ from „A‟ to „B‟ is
called a constant function if and only if the range of „f‟ is a singleton.
Algebraic Function:
A function defined by an algebraic expression is called an algebraic function.
E.g. f(x) =x2+3x+6
Polynomial Function:
A function of the form P(x)=amxn+an–1xn–1+⋯+a1x+a0P(x)=amxn+an–1xn–
1+⋯+a1x+a0
where „n‟ is a positive integer and an,an–1,⋯,a1,a0an,an–1,⋯,a1,a0 are real
numbers is called a polynomial function of degree „n‟.
Linear Function:
A polynomial function with degree „tt‟ is called a linear function. The most general
form of a linear function is
f(x)=ax+bf(x)=ax+b

3. Quadratic Function:
A polynomial function with degree „2‟ is called a quadratic function. The most
general form of a quadratic equation is f(x)=ax2+bx+cf(x)=ax2+bx+c
Cubic Function:
A polynomial function with degree „3‟ is called a cubic function. The most general
form of a cubic function is f(x)=ax3+bx2+cx+df(x)=ax3+bx2+cx+d
Identity Function:
Let f:A→Bf:A→B be a function then „ff‟ is called an identity function
if f(x)=x,∀x∈Af(x)=x,∀x∈A.
Rational Function:
A function R(x)R(x) defined by R(x)=P(x)Q(x)R(x)=P(x)Q(x), where
both P(x)P(x)andQ(x)Q(x) are polynomial functions is called a rational function.
Trigonometric Function:
A function f(x)=sinxf(x)=sin⁡x, f(x)=cosxf(x)=cos⁡x etc., then f(x)f(x) is called a
trigonometric function.
Exponential Function:
A function in which the variable appears as an exponent (power) is called an
exponential function
e.g. (i) f(x)=axf(x)=ax (ii) f(x)=3xf(x)=3x.
Concept of function
Let A and B be any two non–empty sets. Then a function „ff‟ is a rule or law which
associates each element of „A‟ to a unique element of set „B‟.
Notation:
(i) A function is usually denoted by small letters, i.e. f,g,hf,g,h etc. and Greek
letters, i.e. α,β,γ,ϕ ,ψα,β,γ,ϕ ,ψetc.
(ii) If „ff‟ is a function from „A‟ to „B‟ then we write f:A→Bf:A→B.

4. Examples of Functions
Example:1
Find the range of the function f(x)=x+1x–1f(x)=x+1x–1.
Solution:
We have
f(x)=x+1x–1f(x)=x+1x–1
Put x=1x=1
f(1)=1+10=∞f(1)=1+10=∞
Thus, the domain is ∀x∈R–{1}.
Now for the range, we have
f(x)=x+1x–1∴ y=x+1x–1⇒xy–y=x+1⇒xy–x=y+1x=y+1y–1f(x)=x+1x–1∴ y=x+1x–1⇒xy–
y=x+1⇒xy–x=y+1x=y+1y–1
For y=1y=1
x=1+10=∞x=1+10=∞
So, the range of the function ffis {y:y≠1}=]–∞,1[∪]1,∞[{y:y≠1}=]–∞,1[∪]1,∞[.
Example:2
Let f(x)=xx2–16f(x)=xx2–16. Find the domain and range of ff.
Solution:
We have
f(x)=xx2–16f(x)=xx2–16
For x=4x=4
f(4)=416–16=∞f(4)=416–16=∞
For x=–4x=–4
f(–4)=–416–16=∞f(–4)=–416–16=∞

5. Thus, the domain is ∀x∈R–{4,–4}∀x∈R–{4,–4}.
Now for the range, we have
f(x)=xx2–16⇒y=xx2–16⇒y(x2–16)=x⇒yx2–x–16y=0⇒x=–(–1)±(–1)2–4(y)(–
16y)−−−−−−−−−−−−−−−√2(y)⇒x=1±1–64y2−−−−−−√2yf(x)=xx2–16⇒y=xx2–16⇒y(x2–
16)=x⇒yx2–x–16y=0⇒x=–(–1)±(–1)2–4(y)(–16y)2(y)⇒x=1±1–64y22y
For y=0y=0
x=1±1+0−−−−√0=∞
Thus, the range of the function is f=R–{0}

6. Trigonometry
Trigonometry is an important branch of mathematics. The word trigonometry
is derived from three Greek
words: tri (three), goni (angles), metron (measurement). It literally means
“measurement of triangles.”
Trigonometry Table
Angles (In Degrees) 0° 30° 45° 60° 90° 180° 270° 360°
Angles (In Radians) 0° π/6 π/4 π/3 π/2 π 3π/2 2π
sin 0 1/2 1/√2 √3/2 1 0 -1 0
cos 1 √3/2 1/√2 1/2 0 -1 0 1
tan 0 1/√3 1 √3 ∞ 0 ∞ 0
cot ∞ √3 1 1/√3 0 ∞ 0 ∞
csc ∞ 2 √2 2/√3 1 ∞ -1 ∞
sec 1 2/√3 √2 2 ∞ -1 ∞ 1
There are six main trigonometric functions:
Sine (sin)
Cosine (cos)
Tangent (tan)
Secant (sec)
Cosecant (csc)
Cotangent (cot)

7. Period of Trigonometric functions
If your trig function is either a tangent or cotangent, then you'll need to divide
pi by the absolute value of your B. Our function, f(x) = 3 sin(4x + 2), is a sine
function, so the period would be 2 pi divided by 4, our B value
Addition Formulas
The first two addition formulae: sin(A ± B)
This is called an addition formula because of the sum A + B appearing
the formula. Note that it enables us to express the sine of the sum of two
angles in terms of the sines and cosines of the individual angles.

8. The Law of Cosines
The Law of Cosines is useful for finding: the third side of a triangle when we know
two sides and the angle between them (like the example above) the angles of a
triangle when we know all three sides (as in the following example)
The Law of Sines
The Law of Sines. The Law of Sines (or Sine Rule) is very useful for solving
triangles: a sin A = b sin B = c sin C. It works for any triangle:
Algebraic Function
In mathematics, an algebraic function is a function that can be defined as the
root of a polynomial equation. Quite often algebraic functions are algebraic ...

9. Exponential Function
In mathematics, an exponential function is a function of the form. where b is a
positive real number, and in which the argument x occurs as an exponent. For
real numbers c and d, a function of the form is also an exponential function
Logarithms Function
In mathematics, the logarithm is the inverse function to exponentiation. That
means the logarithm of a given number x is the exponent to which another fixed
number, the base b, must be raised, to produce that number x.
Application of Function
In mathematics, function application is the act of applying a function to an
argument from its domain so as to obtain the corresponding value from its range.

10. Concept and Definition of Limit.
Limit (mathematics) ... In mathematics, a limit is the value that a function (or
sequence) "approaches" as the input (or index) "approaches" some
value. Limits are essential to calculus (and mathematical analysis in general) and
are used to define continuity, derivatives, and integrals.
Rates of change and Tangents to Curves
The average rate of change of an arbitrary function f on an interval is represented
geometrically by the slope of the secant line to the graph of f. The
instantaneous rate of change of f at a particular point is represented by the slope
of the tangent line to the graph of f at that point.
Average and Instantaneous Speed
It is different from average speed because average speed is measured by the
total time of a journey divided by the total distance. In contrast, instantaneous
speed measures the smallest interval possible divided by the time it took to move
that distance.
Limit of a function and Limit Laws
If the values of f(x) increase without bound as the values of x (where x≠a)
approach the number a, then we say that the limit as x approaches a is positive
infinity and we write limx→af(x)=+∞. ... For the limit of a function f(x) to exist at a,
it must approach a real number L as x approaches a.

11. Techniques for finding limits
The first technique for algebraically solving for a limit is to plug the number
that x is approaching into the function. If you get an undefined value (0 in the
denominator), you must move on to another technique. But if your function is
continuous at that x value, you will get a value, and you‟re done; you‟ve found
your limit! For example, with this method you can find this limit:
The limit is 3, because f(5) = 3 and this function is continuous at x = 5.
Limits involving infinity and Continues Function
A function is said to be continuous if it is continuous at all points. We say that
the limit of f (x) as x approaches positive infinity is L and write, ... We say that
the limit of f (x) as x approaches negative infinity is L and write, if for any e > 0
there exists N > 0 such that | f (x) - L | < e for all x < -N (e).