For the real valued functions f(x) - |x| - 4 and g(x) = sqrt(5 - x) Find the composition of f o g and the domain in interval notation.
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To find the composition of two functions, we need to substitute the second function into the first function. In this case, we need to find (f o g)(x), where f(x) = -|x| - 4 and g(x) = sqrt(5 - x).
To find (f o g)(x), we substitute g(x) into f(x):
(f o g)(x) = f(g(x))
First, we substitute g(x) into f(x):
f(g(x)) = -|g(x)| - 4
Now, we substitute g(x) = sqrt(5 - x) into f(g(x)):
f(g(x)) = -|sqrt(5 - x)| - 4
To determine the domain of the composition, we need to consider the individual domains of both f(x) and g(x) and their restrictions.
For f(x) = -|x| - 4, there are no specific restrictions on the domain. The domain of f(x) is all real numbers.
For g(x) = sqrt(5 - x), we need to consider the domain of the square root function. The radicand (5 - x) must be greater than or equal to 0 to have a real square root. So, 5 - x ≥ 0.
Solving the inequality, we have:
5 - x ≥ 0
x ≤ 5
Therefore, the domain of g(x) is x ≤ 5.
Now, combining the domains of f(x) and g(x), we find the intersection of their domains:
Domain of (f o g) = Domain of f ∩ Domain of g
Domain of (f o g) = (-∞, ∞) ∩ (-∞, 5] = (-∞, 5]
Thus, the domain of the composition (f o g)(x) is (-∞, 5].
Therefore, the composition is:
(f o g)(x) = -|sqrt(5 - x)| - 4 with the domain (-∞, 5].