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The form of “function as a rule” suggests that we may think of carrying out arithmetic operations like addition, multiplication etc with two functions. If we limit ourselves to real function, then we can attach meaning to equivalent of arithmetic operations with predictable domain intervals. We should, however, clearly understand that function operations with real functions involve new domain for the resulting function. In general, function operation results in contraction of intervals in which new rule formed from algebraic operation is valid.
As pointed out, function operations are defined for a new domain, depending on the type of operations - we carry out. In the nutshell, we may keep following aspects in mind, while describing function operations :
The function operations, like addition, involve more than one function. Each function has its domain in which it yields real values. The resulting domain will depend on the way the domain intervals of two or more functions interact. In order to understand the process, let us consider two functions “ ${y}_{1}$ ” and “ ${y}_{2}$ ” as given below.
$${y}_{1}=\sqrt{\left({x}^{2}-3x+2\right)}$$
$${y}_{2}=\frac{1}{\sqrt{\left({x}^{2}-3x-4\right)}}$$
Let " ${D}_{1}$ " and" ${D}_{2}$ " be their respective domains. Now, the expressions in the square roots need to be non-negative. For the first function :
$${x}^{2}-3x+2\ge 0$$
$$\Rightarrow \left(x-1\right)\left(x-2\right)\ge 0$$
The sign diagram is shown here. The domain for the function is the intervals in which function value is non-negative.
$$\Rightarrow {D}_{1}=-\infty <x\le 1\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}2\le x<\infty $$
Note that domain, here, includes end points as equality is permitted by the inequality "greater than or equal to" inequality. In the case of second function, square root expression is in the denominator. Thus, we exclude end points corresponding to roots of the equation.
$${x}^{2}-3x-4>0$$
$$\Rightarrow \left(x+1\right)\left(x-4\right)>0$$
$$\Rightarrow {D}_{2}=-\infty <x<-1\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}4<x<\infty $$
Now, let us define addition operation for the two functions as,
$$y={y}_{1}+{y}_{2}$$
The domain, in which this new function is defined, is given by the common interval between two domains obtained for the individual functions. Here, domain for each function is shown together one over other for easy comparison.
For new function defined by addition operation, values of x should be such that they simultaneously be in the domains of two functions. Consider for example, x = 0.75. This falls in the domain of first function but not in the domain of second function. It is, therefore, clear that domain of new function is intersection of the domains of individual functions. The resulting domain of the function resulting from addition is shown in the figure.
$$\Rightarrow D={D}_{1}\cap {D}_{2}$$
This illustration shows how domains interact to form a new domain for the new function when two functions are added together.
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