Find the composite functions f @ g and g @ f. Find the domain of each composite function. Are the two composite functions equal f(x) = 3x + 1 g(x) = −x

Find the composite functions f @ g and g @ f. Find the domain of each composite function. Are the two composite functions equal f(x) = 3x + 1 g(x) = −x

Question
Composite functions
asked 2021-02-25
Find the composite functions \(\displaystyle{f}\circ{g}\) and \(\displaystyle{g}\circ{f}\). Find the domain of each composite function. Are the two composite functions equal
f(x) = 3x + 1
g(x) = −x

Answers (1)

2021-02-26
Consider the function:
f(x)=3x+1, g(x)=−x
To calculate \(\displaystyle{\left({f}\circ{g}\right)}\)
\(\displaystyle{\left({f}\circ{g}\right)}{\left({x}\right)}={f{{\left({g{{\left({x}\right)}}}\right)}}}\)
=3(g(x))+1
=3(−x)+1
=−3x+1
To calculate \(\displaystyle{\left({g}\circ{f}\right)}\)
\(\displaystyle{\left({g}\circ{f}\right)}{\left({x}\right)}={g{{\left({f{{\left({x}\right)}}}\right)}}}\)
=g(3x+1)
=−(3x+1)
=−3x−1
To calculate the domain of \(\displaystyle{\left({f}\circ{g}\right)}\) and \(\displaystyle{\left({g}\circ{f}\right)}\) function, The domain of a function is the set of input values for which the function is real or defined,
Here,The composite function \(\displaystyle{\left({f}\circ{g}\right)}\) and \(\displaystyle{\left({g}\circ{f}\right)}\) has no undefined points.
Therefore the domain of both composite functions is \(\displaystyle-\infty{<}{x}{<}\infty\)</span>
Both composite functions are not equal.
0

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