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Operations on functions
(extra-curricular topic)

The sum of functions is defined as follows.

DEFINITION: If the functions \latex{f:A_{1}\longrightarrow B,g:A_{2}\longrightarrow B} are given then the sum of the two functions is the function \latex{ h } which maps from the set \latex{A_{1}\cap A_{2}} to set \latex{ B } and for any \latex{x\in A_{1}\cap A_{2},}

\latex{h(x)=f(x)+g(x)}.

Function \latex{ h } is often written as \latex{h=f+g.}

For example, the sum of the functions \latex{f:\R\longrightarrow \R,f(x=x)} and \latex{g:\R\ \left\{0\right\} \longrightarrow \R,g(x)=\frac{1}{x}} is \latex{h=f+g:\R\left\{0\right\}\longrightarrow \R,h(x)=x+\frac{1}{x}.}

The product of functions is defined similarly.

DEFINITION: If the functions \latex{f:A_{1}\longrightarrow B,g:A_{2\longrightarrow b}} are given then the product of the two functions is the function \latex{ h } which maps from the set \latex{A_{1}\cap A_{2}} \latex{ 2 } to set \latex{ B } and for any \latex{x\in A_{1}\cap A_{2},}

\latex{h(x)=f(x)\times g(x).}

Function h is often written as \latex{h=f\times g.}

For example, the product of the functions \latex{f: \R\longrightarrow \R, f(x)=2x} and \latex{g:\R\longrightarrow \R,g(x)=\frac{1}{1+x^{2} }} is \latex{h:\R\longrightarrow \R,h(x)=\frac{2x}{1+x^{2} }.}

It is unnecessary to define the difference of two functions since the function \latex{f – g} can be viewed as sum of functions f and \latex{(-1)\times g} where for the sake of brevity we denoted by \latex{–1} the constant function \latex{h:\R\longrightarrow \R,h(x)=-1.}

Similarly it is worth defining the quotient of two functions as the product of the dividend and the reciprocal of the divisor. This way it is enough to define the reciprocal of a function.

DEFINITION:The reciprocal of a function \latex{f:A\longrightarrow B} is the function \latex{g:A_{1}\longrightarrow B } which satisfies that \latex{x\in A_{1}} if and only if \latex{x\in A} and \latex{f(x)\neq 0} and additionally for any \latex{x\in A_{1}} it follows that

\latex{g(x)=\frac{1}{f(x)}.}
The reciprocal of function f is often written as \latex{\frac{1}{f}.}

For example, the reciprocal of the function \latex{f:\R\longrightarrow \R, f(x)=x^{2}} is the function \latex{g:\R \left\{0\right\}\longrightarrow \R, g(x)=\frac{1}{x^{2} }.}

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The operations above are simply the equivalents of the operations between the images, that is, between real numbers.

The composition of functions, on the other hand, is an operation that only exists among functions. The sets in the following definitions can be not only number sets but arbitrary nonempty sets.

DEFINITION:Let the functions \latex{f:A\longrightarrow B} and \latex{g:C\longrightarrow D} be given and suppose that \latex{D\subseteq A.} Then the function \latex{h:C\longrightarrow B} for which for any \latex{x\in C} it follows that \latex{h(x)=f(g(x))} is called the composite function of functions \latex{ f } and \latex{ g } (in that order). It is commonly denoted by

\latex{h=f\circ g.}

It worth noting that for the existence of the function \latex{g\circ f} the necessary condition is \latex{B\subseteq C} and in this case it is defined by \latex{g\circ f:A\longrightarrow D,g\circ f(x)=g(f(x)).}

For example if \latex{f:\R\longrightarrow \R,f(x)=\sin x} and \latex{g:\R\longrightarrow \R,g(x)=x^{2}-x+1,} then since \latex{R_{g}\subseteq D_{f}} and \latex{R_{f}\subseteq D_{g},} both \latex{f\circ g} and \latex{g\circ f} exist and

\latex{f\circ g=\sin (x^{2}-x+1 ),}
\latex{g\circ f=(\sin x)^{2}-\sin x+1.}

Thus the operation \latex{\circ} for functions is non-commutative, usually \latex{f\circ g\neq g\circ f.}
Although it is possible to show that the operation \latex{\circ} is associative, that is, if the appropriate sets meet the criteria, then \latex{(f\circ g)\circ h=f\circ (g\circ h).}

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In the definition of functions mapping (associating) is a one way connection. In a certain family of functions this connection can be inverted, thus leading to the concept of the inverse function. Again, we are discussing arbitrary (not necessarily number-number) functions.

DEFINITION: Suppose that the range of the function \latex{f:A\longrightarrow B} is \latex{ B } and the correspondence is one-to-one, that is, if \latex{x_{1}\neq x_{2},x_{1},x_{2}\in A} then \latex{f(x_{1} )\neq f(x_{2} )} (it is a bijection). Then the function \latex{ g }, which is defined as follows:
\latex{g:B\longrightarrow A} and for any \latex{b\in B} it follows that \latex{g(b)=a} if and only if \latex{f(a)=b,}
is called the inverse function of \latex{ f }, and it is denoted by \latex{g=f^{-1}.}

Examples for inverse functions

Let \latex{ S } be the points of the plane and choose a line \latex{ e }. Let \latex{ f } be the function which maps \latex{ S } to \latex{ S } using the following command: for every \latex{P\in S} let \latex{f(P)} be its reflection through the axis \latex{ e }. Since reflections are one-on-one correspondences, \latex{ f } has an inverse and \latex{f^{-1}=f} because the mirror image of a point's mirror image is the point itself. It can be said that \latex{f\circ f} is the identical mapping which maps every point to itself.
Let \latex{f:\R\longrightarrow \R^{+}, f(x)=2^{x},} then since we know that \latex{ f } is strictly increasing, the correspondence is one-on-one.

Its inverse function is (Figure 3):

\latex{f^{-1}:\R^{+}\longrightarrow R,f^{-1}(x)=\log _{2}x.}

If \latex{f:R_{0}^{+}\longrightarrow R_{0}^{+}, f(x)=x^{2}} then \latex{ f } is strictly increasing, therefore it is a one-on-one correspondence.

The inverse of \latex{ f } is

\latex{f^{-1}:\R_{0}^{+}\longrightarrow \R_{0}^{+},f^{-1}(x)=\sqrt{x}.}

It follows from the definition of the inverse function that if \latex{ f } is a number-to-number function, then the point \latex{(x:f(x))} is on the graph of \latex{ f } if and only if the point \latex{(f(x);x)} is on the graph of the inverse function \latex{f^{-1}.} Since these two points are mirror images of each other with the line \latex{y=x} as axis, the graph of the inverse function \latex{f^{-1}} can be found by reflecting the graph of \latex{ f } through the line \latex{y = x.} (Figure 4)

Exercises

{{exercise_number}}. The following two functions are given: \latex{f:\R\longrightarrow \R, f(x)=x^{2}} and \latex{g:\R\longrightarrow \R, g(x)=2^{x}.} Find the following functions:

\latex{f\circ f;}

\latex{f\circ g;}

\latex{g\circ f;}

\latex{g\circ g.}

{{exercise_number}}. Let the function \latex{f:\R\longrightarrow \R,f(x)=\frac{x}{\sqrt{1+x^{2} } }} be given. Find the functions \latex{f\circ f,f\circ f\circ f.}

What is the function \latex{f\circ f\circ f...\circ f} (where there are \latex{ n } copies of \latex{ f } in the formula)?

{{exercise_number}}. Find the inverses of the following functions:

\latex{f:\R\longrightarrow \R f(x)=2x+3;}

\latex{g:\R \left\{-1\right\}\longrightarrow \R \left\{-1\right\}, g(x)=\frac{1-x}{1+x}; }

\latex{h:\left[0;1\right]\longrightarrow \left[0;1\right], h(x)=\sqrt{1-x^{2} ;}}

\latex{k:\left[-1;0\right]\longrightarrow \left[0;1\right],k(x)=\sqrt{1-x^{2} }. }

Mathematics 12.

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