Considere as fun\u00e7\u00f5es reais de vari\u00e1vel real assim definidas: \\[\\begin{matrix}
\nf:x\\to {{(\\sqrt{x}+3)}^{2}} & \\text{e} & g:x\\to {{(\\sqrt{x}-3)}^{2}}\u00a0 \\\\
\n\\end{matrix}\\]<\/p>\n
- \n
- Determine o dom\u00ednio de f e de g.<\/li>\n
- Determine, se existirem, os zeros de f e de g.<\/li>\n
- Caracterize as fun\u00e7\u00f5es $(f+g)$ e\u00a0 $(f\\times g)$ e apresente as express\u00f5es de $(f+g)(x)$ e $(f\\times g)(x)$ na forma mais simplificada poss\u00edvel.<\/li>\n<\/ol>\nResolu\u00e7\u00e3o >><\/a><\/span><\/div><\/div>\n\n\nResolu\u00e7\u00e3o<\/b><\/span><\/p>\n
- \n
- Ora, ${{D}_{f}}={{D}_{g}}=\\left\\{ x\\in \\mathbb{R}:x\\ge 0 \\right\\}=\\mathbb{R}_{0}^{+}$.
\n\u00ad<\/li>\n - Ora,
\n\\[\\begin{array}{*{35}{l}}
\nf(x)=0 & \\Leftrightarrow\u00a0 & {{(\\sqrt{x}+3)}^{2}}=0\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\sqrt{x}+3=0\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & x\\in \\left\\{ {} \\right\\}\u00a0 \\\\
\n\\end{array}\\]<\/p>\nA fun\u00e7\u00e3o f n\u00e3o tem zeros.<\/p>\n
Ora,
\n\\[\\begin{array}{*{35}{l}}
\ng(x)=0 & \\Leftrightarrow\u00a0 & {{(\\sqrt{x}-3)}^{2}}=0\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\sqrt{x}-3=0\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & x=9\u00a0 \\\\
\n\\end{array}\\]<\/p>\nA fun\u00e7\u00e3o g tem apenas um zero: $x=9$.<\/p>\n
\u00a0
![\"\"](\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/11-04-2011-Ecra007-300x225.png\" "\\"Gr\u00e1fico")
<\/a>\u00a0 ![\"\"](\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/11-04-2011-Ecra008-300x225.png\" "\\"Gr\u00e1fico")
<\/a><\/p>\n<\/li>\n- Ora, ${{D}_{f+g}}={{D}_{f}}\\cap {{D}_{g}}=\\mathbb{R}_{0}^{+}$.
\nComo
\n\\[\\begin{array}{*{35}{l}}
\n(f+g)(x) & = & f(x)+g(x)\u00a0 \\\\
\n{} & = & {{(\\sqrt{x}+3)}^{2}}+{{(\\sqrt{x}-3)}^{2}}\u00a0 \\\\
\n{} & = & {{(\\sqrt{x})}^{2}}+6\\sqrt{x}+9+{{(\\sqrt{x})}^{2}}-6\\sqrt{x}+9\u00a0 \\\\
\n{} & = & \\left| x \\right|+9+\\left| x \\right|+9\u00a0 \\\\
\n{} & = & 2x+18\\,\\,(\\text{pois }x\\in \\mathbb{R}_{0}^{+})\u00a0 \\\\
\n\\end{array}\\]
\nent\u00e3o,
\n\\[\\begin{array}{*{35}{l}}
\nf+g: & \\mathbb{R}_{0}^{+}\\to \\mathbb{R}\u00a0 \\\\
\n{} & x\\to 2x+18\u00a0 \\\\
\n\\end{array}\\]Ora, ${{D}_{f\\times g}}={{D}_{f}}\\cap {{D}_{g}}=\\mathbb{R}_{0}^{+}$.
\nComo
\n\\[\\begin{array}{*{35}{l}}
\n(f\\times g)(x) & = & f(x)\\times g(x)\u00a0 \\\\
\n{} & = & {{(\\sqrt{x}+3)}^{2}}.{{(\\sqrt{x}-3)}^{2}}\u00a0 \\\\
\n{} & = & {{\\left[ (\\sqrt{x}+3).(\\sqrt{x}-3) \\right]}^{2}}\u00a0 \\\\
\n{} & = & {{\\left[ {{(\\sqrt{x})}^{2}}-9 \\right]}^{2}}\u00a0 \\\\
\n{} & = & {{x}^{2}}-18\\left| x \\right|+81\u00a0 \\\\
\n{} & = & {{x}^{2}}-18x+81\\,\\,(\\text{pois }x\\in \\mathbb{R}_{0}^{+})\u00a0 \\\\
\n\\end{array}\\]
\nent\u00e3o
\n\\[\\begin{array}{*{35}{l}}
\nf\\times g: & \\mathbb{R}_{0}^{+}\\to \\mathbb{R}\u00a0 \\\\
\n{} & x\\to {{x}^{2}}-18x+81\u00a0 \\\\
\n\\end{array}\\]<\/p>\n - Ora, ${{D}_{f+g}}={{D}_{f}}\\cap {{D}_{g}}=\\mathbb{R}_{0}^{+}$.
- Ora, ${{D}_{f}}={{D}_{g}}=\\left\\{ x\\in \\mathbb{R}:x\\ge 0 \\right\\}=\\mathbb{R}_{0}^{+}$.
![\"\"](\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/11-04-2011-Ecra009-300x225.png\" "\\"Gr\u00e1fico")
<\/a>\u00a0 ![\"\"](\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/04\/11-04-2011-Ecra010-300x225.png\" "\\"Gr\u00e1fico")
<\/a><\/p>\n<\/li>\n<\/ol>\n