Consider the following definitions of operations in functions:
(f + g)(x) = f(x) + g(x) (f - g)(x) = f(x) - g(x) (fg)(x) = f(x)g(x) and (f/g)(x) = f(x)/g(x)consider, f(x) = $\sqrt{x}$ and g(x) = $\sqrt{1-x}$,
here, the domain of f(x) = [0, ∞), and the range is also the same,
for g(x) the domain is [- ∞, 1] and the range is [0, ∞).
Now, consider the following operations from this function:
(f+g)(x)--------------> $\sqrt{x}$ + $\sqrt{1 -x}$(domain: [0,1]
(f-g)(x)--------------> $\sqrt{x}$ - $\sqrt{1- x}$(domain: [0,1]
(f.g)(x)--------------> $\sqrt{x}$.$\sqrt{1-x}$---->$\sqrt{x(1-x)}$(domain:[0,1]
(f/g)(x)------------>$\sqrt{x/(1-x)}$(domain: [0, 1)
(g/f)(x)------------->$\sqrt{(1-x)/x}$(domain: (0,1]
It is quite obvious for the domain to be the common interval of both the functions, since the operated function(e.g. (h: (f+g)(x)), must satisfy for both the intervals for both the functions.
But, What follows for the range? Below I have presented a graph of the two operations plotted on geogebra:click to enlarge, the equations are on left
Example for the operation (f-g)(x), the range is not [0,∞), nor something related to the union or intersection of the two ranges.
A bit down the list, the more interesting of all is the division, (f/g) omits the right interval of the domain and (g/f) omits the left. Is this a coincidence or a general rule, if so, is there a proof for this?
And, lastly, is there a general rule to guide the values of domain and ranges for such algebraic operations?
$\endgroup$1 Answer
$\begingroup$Consider a function h(z), such that $$h(z)=f(x)/z$$
The domain of z in this case will be:
$$z \in (-\infty,0),(0,\infty) $$ and its range will be the same
When we substitute a function f or g for z, the range of what we substitute becomes the domain of h. Making the composite function h(g(x)).
But because the domain of h(z) is also the conjunction of f and z's domain (as you observed earlier), our final functions domain will be the conjunction of f(x)'s domain, h(z)'s original domain and the range of what we substituted.
As a general rule, whenever you divide two functions; take the denominators range and form a conjunction of it with the numerators domain, and form a conjunction of that with a composite functions domain, that has the form of your new function.
Symbolically $$dom \ h(g(x))=(range \ g) \cap (dom\ h(z)) \cap (dom\ f)$$
In answer to your other remark:
A bit down the list, the more interesting of all is the division, (f/g) omits the right interval of the domain and (g/f) omits the left. Is this a coincidence or a general rule, if so, is there a proof for this?
no that symmetry is just a coincidence.
Excellent question.
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