To find the domain of the composite function f o g, you need to consider the restrictions imposed by both individual functions, f and g.
The domain of f is given as 4 < x < 8. Since g(x) is defined for 1 < x < 5, the values of x that can be fed into g must be within this range for the composite function f o g to be defined.
Therefore, the domain of f o g is the intersection of the domain of g and the range of values that satisfy the requirement for g, given as 1 < x < 5.
For the range of f o g, you can start by finding the expression for f(g(x)), which you correctly determined as 7 - x.
To find the range, consider the possible values that can be obtained from 7 - x. Since the original function f(x) is defined as 9 - x, the range of f(x) is all real numbers except x = 9. Therefore, when we substitute g(x) = x + 2 into f(g(x)), we need to determine if there are any x values for which 7 - x equals 9.
Setting 7 - x = 9 and solving for x, we get x = -2. So, the only value of x that results in f(g(x)) being undefined is x = -2. Therefore, the range of f o g is all real numbers except x = -2.
In summary:
- The domain of f o g is 1 < x < 5, which is the intersection of the domain of g and the range of valid values for g.
- The range of f o g is all real numbers except x = -2.