Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Different Categories of function

(i) Algebraic function:

Algebraic functions consist of algebraic operations such as addition, subtraction, multiplication, division, square, square root, etc.

Polynomial functions, the rational functions are such type of algebraic function.

(ii) Transcendental function:

The functions which are not algebraic are called the transcendental function.

(a) Trigonometric function:

Sine:  R\rightarrow [-1,1]

Cosine:  R\rightarrow [-1,1]

Tangent:  R^1\rightarrow R  where  R^1=R-\{(2n+1)\dfrac{\bar{a}}{2}: n \in Z\}

Co-tangent:  R^{11}\rightarrow R where R^{11} =R -\{n\bar{n}: n \in Z\}

Secant:  R^1\rightarrow R

CoSecant:  R^{11}\rightarrow R 

The above functions are trigonometric functions.

(b) Inverse Trigonometric function:

\sin^{-1}x , \cos^{-1}x , \tan^{-1}x ,\cot^{-1}x , \sec^{-1}x , \csc^{-1} x are inverse trigonometric function.

(c) Exponential function:

 a^x, a>0 and  a\neq 1, e^x are called exponential function.

x^{(\sin x)}, (\cos x)^{\log x} are all exponential function.

(d) Logarithmic function:

Function  \log_ax, \log_e(1+x) etc. are called logarithmic function.

(e) Irrational Power function:

Irrational powers of positive real numbers such as  x^{\sqrt{2}}, x^{\sqrt{5}} etc. are irrational power functions.

(iii) Odd and even functions:

The function  f is called odd if f(-x)=-f(x) and it is called even if  f(-x)=f(x), \forall x \in D_f.

 \sin(-\theta )=-\sin\theta (\text { odd function })

 \cos(-\theta )=\cos\theta (\text { even function })

 \rightarrow If  f is a real function, then

g(x)=\dfrac{f(x)+f(-x)}{2} \text{ is an even function }

and

h(x)=\dfrac{f(x)-f(-x)}{2} \text{ is an odd function }

Since \begin{aligned} f(x)&=g(x)+h(x)\\&=\dfrac{f(x)+f(-x)}{2}+\dfrac{f(x)-f(-x)}{2}\end{aligned}

is written as “every function can be written as the sum of an odd and an even function.”

(iv) Periodic function:

A function  f is called periodic with period K if  f(x+k)=f(x) for some constant  k \neq 0 .

The least positive value of k for which  f(x+k)=f(x) holds is called the fundamental period of  f .

 

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