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Solución
Given the functions:
Let’s find both compositions
and
, along with their domains and ranges.
1. Composition
Definition:
Calculation:
Domain:
Both and
are polynomial functions, which are defined for all real numbers. Therefore, the domain of
is:
Both
Range:
Thus, the range is:
2. Composition
Definition:
Calculation:
Expanding the square:
This can also be written in vertex form by completing the square:
Domain:
Again, since both and
are defined for all real numbers, the domain of
is:
Again, since both
Range:
Thus, the range is:
Summary
-
(x): -
(x):
Final Answer:
After simplifying, the compositions are
f ∘ g (x ) = 6x² – 1 with domain all real numbers and range [ –1, ∞ )
g ∘ f (x ) = 12x² – 12x + 3 with domain all real numbers and range [ 0, ∞ )
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Beyond the Answer
To find
and
:
-
For
: The domain ofis all real numbers , since a quadratic function is defined for all . The range of is all real numbers because the minimum value of is 0 (when ). -
For
: Here, the domain ofis also all real numbers . For the range of , this is a quadratic that opens upwards, so it has a minimum value. Completing the square shows that the minimum occurs at , giving the minimum value of . Thus, the range is .
So, we have:
-
; Domain: , Range: -
; Domain: , Range:

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