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

Equality of functions

If  f_1   and   f_2  are two relations on  X    to  Y , then they are called equal if they are equal as subsets of  X \times Y  .

When applied to functions, this means f_1,f_2: X\rightarrow Y    are equal  (f_1=f_2)   if    \{(x,f_1(x)): x \in X\}= \{(x,f_2(x)):x \in X\}  i.e.,  f_1(x)=f_2(x) , for each  x \in X  .

 \rightarrow  \text { dom } f_1 = \text { dom }f_2 & codomain of  f_1= codomain of f_2 .

Some operations in functions:

Let f,g : X\rightarrow R then

(I) The addition of two functions,i.e., f+g defined by

(f+g)(x)=f(x)+g(x), (x \in X)

(ii) The multiplication f(g) is defined by

 (fg)(x)= f(x) g(x), (x \in X)

(iii) The quotient \dfrac{f}{g} is defined by

\left ( \dfrac{f}{g} \right )(x)= \dfrac{f(x)}{g(x)}, x \in X where  g(x)\neq 0

(iv) The subtraction of the two functions,

 (f-g)(x)=f(x)-g(x), (x \in X)

Example:

f(x)= | x |, \forall x \in R, g(x)=x,\forall x \in R

(i) (f+g)(x)=f(x)+g(x)=\begin{Bmatrix} 2x, & \text { if } & x \geq 0 \\ 0 & \text { if } & n < 0 \end{Bmatrix}

(ii) (f-g)(x)=f(x)-g(x)=\begin{Bmatrix} 0 & \text { if } & x \geq 0 \\ -2x & \text { if } & n < 0 \end{Bmatrix}

(iii) (fg)(x)=f(x)g(x)=\begin{Bmatrix} x^2 & \text { if } & x \geq 0 \\ -x^2 & \text { if } & n < 0 \end{Bmatrix}

(iv) \left ( \dfrac{f}{g} \right )(x)=\dfrac{f(x)}{g(x)}=\begin{Bmatrix} 1 & \text { if } & x > 0 \\ -1 & \text { if } & n < 0 \end{Bmatrix}

 \left ( \dfrac{f}{g} \right )(x) is not defined x=0 .

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