Compose the following function and state the domain of the composed function. f(x)= 1/x-1 g(x)= sqrt(1-x^2) a) g(f(-2))

Yulia 2021-01-10 Answered
Compose the following function and state the domain of the composed function.
f(x)=1x1
g(x)=1x2
a) g(f(2))
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Expert Answer

Alannej
Answered 2021-01-11 Author has 104 answers
To find g(f(−2)), substitute -2 for x in the function f(x)=1x1 and then 13 for x in the function g(x)=1x2, and simplify.
g(f(2))=g(121)
=1(13)2
=119
=919
=83
=223
Therefore, g(f(2))=223
To obtain the composite function (gf)(x), substitute f(x)=1x1 and g(x)=1x2 in the definition of composite function (gf)(x)=g[f(x)], and simplify.
(gf)(x)=g[f(x)]
=g|1x1|
=1(1x1)2
=(x1)21(x1)2
=(x22x+11x1
=x(x2)x1
Therefore, g[f(x)]=x(x2)x1
To determine the domain of the composite function (gf)(x), use the definition of the composite function g[f(x)]=x(x2)x1 and identify the values of x, for which the denominator is not equal to zero and radicand is positive.
x0
x20
x2
x10
x1
Hence, the domain of the composite function g[f(x)]=x(x2)x1 is
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