Functions - Composite Functions
Functions - Composite Functions
\(\mathrm{f}:x\mapsto -{{e}^{\left| 3-x \right|}},x\in \mathbb{R}\) and \(\mathrm{g}:x\mapsto -1-{{\left( x-1 \right)}^{2}},x\in \mathbb{R}\).
Determine, with reasons, whether \(\mathrm{fg}\) and \(\mathrm{gf}\) exist. If the composite function exists, give the rule, domain and range in exact form.
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Answer: \(\text{fg}\) and \(\text{gf}\) exist. \(\text{fg}\hspace{0.1em}\left( x \right)=-{{e}^{\left| 4+{{\left( x-1 \right)}^{2}} \right|}},{{\text{D}}_{\text{fg}}}=(-\infty ,\infty ),{{R}_{\text{fg}}}=(-\infty ,-{{e}^{4}}]\); \(\text{gf}\hspace{0.1em}\left( x \right)=-1-{{\left( {{e}^{\left| 3-x \right|}}+1 \right)}^{2}},{{D}_{\text{gf}}}=(-\infty ,\infty ),{{R}_{\text{gf}}}=(-\infty ,-5]\)
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