Let f and g be defined by
f(x) = 9-x such that 4<x<8
g(x) = x+2 such that 1<x<5
[the "<" signs are less than or equal to]
Find the domains of both f o g and g o f algebraically. Find the range of f o g algebraically.
I got the equation for f(g(x)), which is 7-x. But how do i find the domains and range for these two composite functions?
Thanks!
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.
To find the domain of the composite function f o g or g o f, we need to consider the domain restrictions of both functions involved.
1. Domain of f o g (f composed with g):
To find the domain of f o g, we need to consider the domain of g(x) and the values of x that would satisfy the domain restrictions of f(x).
The domain of g(x) is given as 1 < x < 5. This means that x can take any value between 1 and 5, excluding 1 and 5 themselves.
Next, we need to consider the domain restrictions of f(x), which are given as 4 < x < 8. This means that x can take any value between 4 and 8, excluding 4 and 8 themselves.
Since g(x) is restricted to 1 < x < 5, and f(x) is restricted to 4 < x < 8, the composite function f o g will only be defined for values of x that satisfy both conditions.
Therefore, the domain of f o g is the intersection of the respective domains, which is 4 < x < 5.
2. Domain of g o f (g composed with f):
To find the domain of g o f, we again need to consider the domain of f(x) and the values of x that would satisfy the domain restrictions of g(x).
The domain of f(x) is given as 4 < x < 8, as mentioned earlier.
Next, we need to consider the domain restrictions of g(x), which are given as 1 < x < 5, as mentioned earlier.
Since f(x) is restricted to 4 < x < 8, and g(x) is restricted to 1 < x < 5, the composite function g o f will only be defined for values of x that satisfy both conditions.
Therefore, the domain of g o f is the intersection of the respective domains, which is 4 < x < 5.
3. Range of f o g:
To find the range of f o g, we need to substitute the expression f(g(x)) = 7 - x into the domain of f o g, which we found to be 4 < x < 5.
When we substitute x = 4 into f(g(x)), we get f(g(4)) = 7 - 4 = 3.
And when we substitute x = 5 into f(g(x)), we get f(g(5)) = 7 - 5 = 2.
Therefore, the range of f o g is 2 < f(g(x)) < 3.
Note: It is important to carefully consider the domains and range for composite functions to ensure that the functions are well-defined and avoid any undefined or imaginary values.