The domain of the operation of two functions f and g is said to be the intersection of the individual domains, which is true for addition, subtraction, and multiplication of functions.
For the division of functions, an extra condition must be true: the denominator cannot be 0.
For example:
[tex]\begin{gathered} f(x)=\sqrt{x} \\ g(x)=\sqrt{x-4} \end{gathered}[/tex]
The domain of f is x ≥ 0 and the domain of g is x ≥ 4.
Note the intersection of the domains is x ≥ 4.
Now consider the function:
[tex]h(x)=\frac{f(x)}{g(x)}=\frac{\sqrt{x}}{\sqrt{x-4}}[/tex]
Here, we must exclude x = 4 from the domain of h because it would make the denominator 0, thus the domain of h(x) is x > 4 (excluding x = 4).
To find the range, we set x to a value real close to 4 (from the right side), like x = 4.001:
[tex]h(x)=\frac{\sqrt{4.001}}{\sqrt{4.001-4}}=63.2[/tex]
The value of h goes up as the values of x are closer to 4, thus the range has no limit to the right side.
Now give x a very high value like x = 100000:
[tex]h(100000)=\frac{\sqrt{100000}}{\sqrt{100000-4}}=1.00002[/tex]
The value of h goes closer to 1 when x goes up to infinity. Thus the range is
(1, +∞).
In the following set of diagrams, Hanna's explanation is shown:
