F asked in Science & MathematicsMathematics · 2 months ago

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• 2 months ago

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.

• Pope
Lv 7
2 months ago

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]

• 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]