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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