Find the composite functions f @ g and g @ f. Find the domain of each composite function. Are the two composite functions equal? f(x) = x^2 − 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) = x^2 − 1 g(x) = −x

Question
Composite functions
asked 2020-10-18
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?
\(\displaystyle{f{{\left({x}\right)}}}={x}^{{2}}−{1}\)
g(x) = −x

Answers (1)

2020-10-19
To find the composite function, \(\displaystyle{f}\circ{g}\) and \(\displaystyle{g}\circ{f}\) and also their respective domains.
Solution:
The functions are \(\displaystyle{f{{\left({x}\right)}}}={x}^{{2}}−{1}\) and g(x)=−x.ZSK
The function \(\displaystyle{f}\circ{g}\) can be evaluated as,
\(\displaystyle{f}\circ{g}={f{{\left({g{{\left({x}\right)}}}\right)}}}\)
\(\displaystyle={f{{\left(−{x}\right)}}}\)
\(\displaystyle={\left(−{x}\right)}^{{2}}−{1}\)
\(\displaystyle={x}^{{2}}−{1}\)
The function \(\displaystyle{g}\circ{f}\) can be evaluated as,
\(\displaystyle{g}\circ{f}={g{{\left({f{{\left({x}\right)}}}\right)}}}\)
\(\displaystyle={g{{\left({x}^{{2}}−{1}\right)}}}\)
\(\displaystyle=−{\left({x}^{{2}}−{1}\right)}\)
\(\displaystyle=−{x}^{{2}}+{1}\)
The obtained composite function are not equal. And the domain for both the function is complete real numbers as it is defined for each value of x.
0

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