Given f(x) = x^2, g(x) = x + 7 Find (f @ g)(x). Find the domain of (f @ g)(x). Find (g @ f)(x). Find the domain of (g @ f)(x). Find (f @ f)(x). Find the domain of (f @ f)(x). Find (g @ g)(x). Find the domain of (g @ g)(x).

Question
Composite functions
asked 2020-10-21
Given
\(\displaystyle{f{{\left({x}\right)}}}={x}^{{2}},{g{{\left({x}\right)}}}={x}+{7}\)
Find \(\displaystyle{\left({f}\circ{g}\right)}{\left({x}\right)}.\)
Find the domain of \(\displaystyle{\left({f}\circ{g}\right)}{\left({x}\right)}.\)
Find \(\displaystyle{\left({g}\circ{f}\right)}{\left({x}\right)}.\)
Find the domain of \(\displaystyle{\left({g}\circ{f}\right)}{\left({x}\right)}.\)
Find \(\displaystyle{\left({f}\circ{f}\right)}{\left({x}\right)}.\)
Find the domain of \(\displaystyle{\left({f}\circ{f}\right)}{\left({x}\right)}.\)
Find \(\displaystyle{\left({g}\circ{g}\right)}{\left({x}\right)}.\)
Find the domain of \(\displaystyle{\left({g}\circ{g}\right)}{\left({x}\right)}.\)

Answers (1)

2020-10-22
\(\displaystyle{f}\circ{g{{\left({x}\right)}}}={f{{\left({g{{\left({x}\right)}}}\right)}}}={\left({g{{\left({x}\right)}}}\right)}^{{2}}={\left({x}+{7}\right)}^{{2}}\)
\(\displaystyle={x}^{{2}}+{14}{x}+{49}\)
It is a polunomial so domain is all real numbers
Domain of \(\displaystyle{f}\circ{g{{\left({x}\right)}}}\) is \(\displaystyle{\left(-\infty,\infty\right)}\)
\(\displaystyle{g}\circ{f{{\left({x}\right)}}}={f{{\left({x}\right)}}}+{7}={x}^{{2}}+{7}\)
It is a polunomial so domain is all real numbers
Domain of \(\displaystyle{g}\circ{f{{\left({x}\right)}}}\) is \(\displaystyle{\left(-\infty,\infty\right)}\)
\(\displaystyle{f}\circ{f{{\left({x}\right)}}}={\left({f{{\left({x}\right)}}}\right)}^{{2}}={\left({x}^{{2}}\right)}^{{2}}={x}^{{4}}\)
It is a polunomial so domain is all real numbers
Domain of \(\displaystyle{f}\circ{f{{\left({x}\right)}}}\) is \(\displaystyle{\left(-\infty,\infty\right)}\)
\(\displaystyle{g}\circ{g{{\left({x}\right)}}}={g{{\left({x}\right)}}}+{7}={x}+{7}+{7}={x}+{14}\)
It is a polunomial so domain is all real numbers
Domain of \(\displaystyle{g}\circ{g{{\left({x}\right)}}}\) is \(\displaystyle{\left(-\infty,\infty\right)}\)
0

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