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• In order to be able to take the composition of f with g,  the range of g has to be a subset of the domain of f.

In your problem,  the range of g is (-∞, 1).  the domain of f is (-2, ∞),

(-∞, 1) is not a subset of (-2, ∞) so it's not possible to take the composition of f with g.

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• The function h is a composite, where g is the interior function. It cannot be defined on any x where g(x) is undefined. The domain of the composite function must be a subset of the domain of g. This condition must be satisfied:

x ≥ -2

Also, if x is on the domain of the composite, then g(x) must be on the domain of f. That gives us a second condition:

g(x) ≥ -3

From the graph of y = g(x) we see the that g(x) ≥ -3 only where x ≤ 2.

Combine those two conditions for the domain of the composite.

x ≥ -2 and x ≤ 2

-2 ≤ x ≤ 2

Domain of h: [-2, 2]

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• Anonymous
2 months ago

It's the intersection of the domain of f(x) and the range of g(x)

Domain of f(x) = [-3,infinity)

Range of g(x) = [1, -infinity)

They intersect over the range of [-3,1]

Thus domain of f(g(x)) = [-3,1]

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• Given that you have h(x) = f(g(x)), the output (range) of g(x) is the input (domain) of f(g(x)) and hence the domain of h(x).  But, only the overlap of the range of g(x) and the domain of f(x).  The range of g(x) = (-∞,1].  The domain of f(x) = [-3,∞).  Now to find the domain of f(g(x)), find the overlap of (-∞,1] and [-3,∞) which is [-3,1].

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