Find the domain of the composite function, hog
H(x)=√x , g(x)=-x+3
hog = h(g(x)) = h(-x+3) = √(-x+3)
The term in the square root cannot take negative values. It must be greater than (or equal to) zero.
-x + 3 >= 0
-x >= -3
x =< 3
Hence, the domain of x is given by (-∞, 3]
To find the domain of the composite function h o g, we need to determine the values of x for which the composite function h(g(x)) is defined.
First, let's find h(g(x)). Since h(x) = √x and g(x) = -x + 3, we can substitute g(x) into h(x) to get:
h(g(x)) = √(-x + 3)
In order for √(-x + 3) to be defined, the expression inside the square root must be non-negative. Therefore, we need to solve the inequality -x + 3 ≥ 0.
Adding x to both sides of the inequality gives us:
3 ≥ x
So, the domain of the composite function h o g is all the values of x such that x ≤ 3. Thus, the domain of the composite function h o g is (-∞, 3].
To find the domain of the composite function h o g, we need to consider the domain of the function g and the domain of the function h.
The domain of the function g, denoted as dom(g), is the set of all possible input values for g(x) to be defined. In this case, g(x) = -x + 3.
Since -x + 3 is a linear function, it is defined for all real numbers. Therefore, the domain of g is the set of all real numbers, which can be expressed as:
dom(g) = (-∞, ∞)
The domain of the function h, denoted as dom(h), is the set of all possible input values for h(x) to be defined. In this case, h(x) = √x.
The square root function is defined only for non-negative real numbers. In other words, x must be greater than or equal to zero for h(x) to be defined. Therefore, the domain of h is:
dom(h) = [0, ∞)
Now, we can determine the domain of the composite function h o g by considering the intersection of the domains of h and g.
Taking the intersection of dom(h) and dom(g), we have:
dom(h o g) = dom(h) ∩ dom(g)
= [0, ∞) ∩ (-∞, ∞)
The intersection of these two sets is the set [0, ∞), as all the elements in [0, ∞) are also in (-∞, ∞). Therefore, the domain of the composite function h o g is:
dom(h o g) = [0, ∞)