Given f(x) = 2 - x2, g(x) = sqrt(x + 2) f(x) = sqrt(x), g(x) = sqrt(1 - x) (a) write formulas for f@g and g@f and find the (b) domain and (c) range of each.

Nann 2020-11-26 Answered
Given
f(x)=2x2,g(x)=x+2
f(x)=x,g(x)=1x
(a) write formulas for fg and gf and find the
(b) domain and
(c) range of each.
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Expert Answer

brawnyN
Answered 2020-11-27 Author has 91 answers

(a) fg(x)
f(x+2)
2(x+2)2
2(x+2)
x
gf(X)
g(2x2)
2x2+2
4x2
(b) Now, find the domain of the both composite functions.
The domain of the function f(x) consists of all real and the domain of the function g(x) is [2,).
Now check the domain of the composite function fg (x) = -x. The domain of this function consists of all real values.
To find the domain of fg (x), there are some restrictions. But [2,) is a proper subset of the domain of fg (x).
Therefore, the domain of fg (x) will be [2,).
Similarly, check the domain of the composite function gf (x). The domain of this function is [-2, 2].
To find the domain of gf (x), there are some restrictions. But [-2, 2] is a proper subset of the domain of gf (x).
Therefore, the domain of gf (x) will be [-2, 2].
(c) Now, find the range of the both composite functions.
The range of the composite function fg (x) will be all real values.
i.e. Range = (,)
The range of the composite function gf (x) will be 0 to 2.
i.e. Range = [0, 2].

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