6.0 Operations on Real Functions

1. Addition:

Let $f:{D_1} \to R$ and $g:{D_2} \to R$ be two real functions. Then, their sum $f + g$ is defined as that function from ${D_1} \cap {D_2}$ to $R$ which associates each $x \in {D_1} \cap {D_2}$ to the number $f(x) + g(x)$. $$(f + g)(x) = f(x) + g(x)\;\;\;\forall \;\;x \in {D_1} \cap {D_2}$$

Let $f:{D_1} \to R$ and $g:{D_2} \to R$ be two real functions. Then, their product $fg$ is the function from ${D_1} \cap {D_2}$ to $R$, and is defined as $$(fg)(x) = f(x)g(x)\;\;\;\forall \;\;x \in {D_1} \cap {D_2}$$

Let $f:{D_1} \to R$ and $g:{D_2} \to R$ be two real functions. Then, the difference of $g$ from $f$ is denoted as $f - g$ and is defined as $$(f - g)(x) = f(x) - g(x)\;\;\;\forall \;\;x \in {D_1} \cap {D_2}$$

4. Quotient: Let $f:{D_1} \to R$ and $g:{D_2} \to R$ be two real functions. Then the quotient of $f$ by $g$ is denoted by ${f \over g}$ and is a function from ${D_1} \cap {D_2} - \{ x:g(x) = 0\} $ to $R$ defined by, $$\left( {{f \over g}} \right)(x) = {{f(x)} \over {g(x)}}\;\;\;\;\forall \;\;x \in {D_1} \cap {D_2} - \{ x:g(x) = 0\} \;$$

Let $f:D \to R$ be a real function and $\alpha $ be a scalar. Then the product $\alpha f$ is a function from $D$ to $R$ and is defined as, $$(\alpha f)(x) = \alpha f(x)\;\;\;\forall \;\;x \in D$$

If $f:D \to R$ is a real function, then its reciprocal function ${1 \over f}$ is a function from $D - \{ x:f(x) = 0\} $ to $R$ and is defined as, $$\left( {{1 \over f}} \right)(x) = {1 \over {f(x)}}\;\;\;\;\forall \;\;x \in D - \{ x:f(x) = 0\} \;$$