6.2 Operations on Functions

Now we will define how to combine functions together to create new functions. Let $f, g$ be functions with domains and ranges in $\mathbb{R}$ .

Definition 6.3 (Basic Arithmetic).

Firstly, it is useful to have a working definition of equality.

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$f=g\iff D:=\operatorname{dom}(f)=\operatorname{dom}(g)\ \text{and}\ f(x)=g(x)% \ \forall x\in D$ .

For simple operations like the sum and product, we define the new domain simply as the common domain of the operands.

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$\operatorname{dom}(f+g)=\operatorname{dom}(f\cdot g)=\operatorname{dom}(f)\cap% \operatorname{dom}(g)$ .
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$(f+g)(x)=f(x)+g(x)$ .
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$(f\cdot g)(x)=f(x)g(x)$ .

In the special case of multiplying by a constant function $c$ given by $x\mapsto k,k\in\mathbb{R}$ ,

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$(c\cdot f)(x)=c(x)f(x)=kf(x)$ .

Using this definition with the constant function $x\mapsto-1$ , we can define the differences of functions.

The reciprocal $1/g$ is defined where $g$ is not zero.

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$\operatorname{dom}\left(\frac{1}{g}\right)=\{x\in\operatorname{dom}(g):g(x)% \neq 0\}$ .
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$\left(\frac{1}{g}\right)(x)=\frac{1}{g(x)}$ .

Using this we can define quotients of functions as a product of a function with a reciprocal.

Definition 6.4 (Composition).

The composition $f\circ g:\operatorname{dom}(g)\to\operatorname{range}(f)$ is defined only where $\operatorname{im}(g)$ overlaps with $\operatorname{dom}(f)$ , so we say:

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$\operatorname{dom}(f\circ g)=\{x\in\operatorname{dom}(g):g(x)\in\operatorname{% dom}(f)\}$ .
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$(f\circ g)(x)=f\big{(}g(x)\big{)}$ .