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 a composite function, you need to consider the domains of the individual functions involved and how they interact with each other.
Let's start with the domain of f o g.
The composition f o g means you first evaluate g(x) and then plug that result into f(x). Since g(x) = x + 2, and the domain of g(x) is 1 < x < 5, we need to check if this domain allows us to evaluate f(g(x)).
To do this, we substitute x + 2 into f(x) and check if it falls within the domain of f(x) = 9 - x.
f(g(x)) = 9 - (x + 2) = 7 - x
The domain of f(g(x)) will depend on the value of x in the domain of g(x). Since the domain of g(x) is 1 < x < 5, let's substitute the boundary values into f(g(x)).
For x = 1: f(g(1)) = 7 - 1 = 6
For x = 5: f(g(5)) = 7 - 5 = 2
So, the domain of f(g(x)) is 2 ≤ f(g(x)) ≤ 6 (assuming we are dealing with real numbers). You can express this domain as an interval: [2, 6].
Now let's find the composition g o f (g(f(x))) and its domain.
The composition g o f means you first evaluate f(x) and then plug that result into g(x). Since f(x) = 9 - x, and the domain of f(x) is 4 < x < 8, we need to check if this domain allows us to evaluate g(f(x)).
To do this, we substitute 9 - x into g(x) and check if it falls within the domain of g(x) = x + 2.
g(f(x)) = 9 - x + 2 = 11 - x
The domain of g(f(x)) will depend on the value of x in the domain of f(x). Since the domain of f(x) is 4 < x < 8, let's substitute the boundary values into g(f(x)).
For x = 4: g(f(4)) = 11 - 4 = 7
For x = 8: g(f(8)) = 11 - 8 = 3
So, the domain of g(f(x)) is 3 ≤ g(f(x)) ≤ 7. This domain can be expressed as an interval: [3, 7].
To find the range of f o g algebraically, we need to consider the range of f(g(x)).
Since we already determined f(g(x)) = 7 - x, the range will be the set of all possible values that 7 - x can take.
Let's look at the boundaries of the domain of g(x): 1 < x < 5.
For x = 1: f(g(1)) = 7 - 1 = 6
For x = 5: f(g(5)) = 7 - 5 = 2
So, the range of f o g is the set of all possible outputs from f(g(x)), which is [2, 6].