Graphically and mathematically find the domain and range of g(f(x)).

Given:

By inspection:

Domain:

or

Range:

Analytical Solution:

so we want to find where:

Solution 1

this is a disjunction ('union", "or", "U")

transformed into equivalent sentences:

Domain:

or

Range: the -5 factor will always yield a negative range: y <= 0.

Solution 2

More generally:

this is an absolute value equality

finding the roots:

or

by inspection, the domain will be:

Range: the -5 factor will always yield a negative range: y <= 0.