Answer:
Domain: Set of All Real Numbers. (-∞, ∞)
Range: (0, ∞)
Step-by-step explanation:
[tex]f(x)=e^{x}\\g(x)=x+6[/tex]
We need to evaluate (gof)(x). (gof)(x) means the composition of function g(x) and f(x). In order to find (gof)(x) replace every occurrence of x in g(x) with the expression of f(x) as shown below:
[tex](gof)(x)=g(f(x))\\\\ =g(e^{x})\\\\ =e^{x}+6[/tex]
We have to find the domain and range of this composite function.
Domain:
The expression [tex]e^{x}+6[/tex] is defined for all Real Numbers. It never gets undefined for any value of x. Hence the Domain is set of All Real Numbers.
Range:
The range of the parent exponential function [tex]e^{x}[/tex] is from 0 to positive infinity. The expression [tex]e^{x}+6[/tex] can be related to [tex]e^{x}[/tex] as a vertical shifted function by 6 units upward. As a result the range will also be shifted up. So the range of the composite function will be from 6 to positive infinity. In interval notation this would be expressed as: (0, ∞)
