Given the real functions below, you should be able to determine their domains.
", "ungrouped_variables": ["out", "inp", "num", "n", "c", "rat", "a", "b", "base", "d"], "rulesets": {}, "parts": [{"scripts": {}, "unitTests": [], "marks": 0, "customMarkingAlgorithm": "", "adaptiveMarkingPenalty": 0, "customName": "", "extendBaseMarkingAlgorithm": true, "type": "gapfill", "prompt": "Give an example of a real number that is in the domain of the function
\n\\[f(x)=\\sqrt{x}.\\]
\n[[0]] $\\in \\text{dom}(f)$
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", "$t> \\var{c[0]}$
", "$t\\le\\var{c[0]}$
", "$t< \\var{c[0]}$
", "$t>0$
", "$t\\ne0$
"], "customName": "", "extendBaseMarkingAlgorithm": true, "type": "1_n_2", "prompt": "Given the function \\[g(t)=\\sqrt{\\simplify{t-{c[0]}}},\\] what do we require of $t$ so that $g(t)$ is defined?
\n", "distractors": ["", "", "", "", "", ""], "matrix": ["1", 0, 0, 0, 0, 0], "shuffleChoices": true, "showCorrectAnswer": true, "useCustomName": false, "variableReplacements": [], "displayType": "radiogroup"}, {"scripts": {}, "displayColumns": "1", "unitTests": [], "customMarkingAlgorithm": "", "adaptiveMarkingPenalty": 0, "minMarks": 0, "maxMarks": 0, "showCellAnswerState": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "marks": 0, "choices": ["$\\{\\simplify{{inp}}\\in\\mathbb{R}:\\,\\simplify{{inp}<={a}} \\text{ or } \\simplify{{inp}>={b}}\\}$
", "$\\{\\simplify{{inp}}\\in\\mathbb{R}:\\,\\simplify{{a}<={inp} <={b}}\\}$
", "$\\{\\simplify{{inp}}\\in\\mathbb{R}:\\,\\simplify{{inp}>={a*b}}\\}$
", "$\\{\\simplify{{inp}}\\in\\mathbb{R}:\\,\\simplify{{inp}>=0}\\}$
", "$\\{\\simplify{{inp}}\\in\\mathbb{R}:\\,\\simplify{{min([a+b,a*b])}<={inp} <={max([a+b,a*b])}}\\}$
"], "customName": "", "extendBaseMarkingAlgorithm": true, "type": "1_n_2", "prompt": "Given the function \\[\\simplify{{out}({inp})}=\\sqrt{\\simplify{{inp}^2-{a+b}{inp}+{a*b}}},\\] which of the following represents the domain of $\\simplify{{out}}$?
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", "$\\{\\simplify{{inp}}\\in\\mathbb{R}:\\,\\simplify{0<={inp} <={d}}\\}$
", "$\\{\\simplify{{inp}}\\in\\mathbb{R}:\\,\\simplify{{inp}>={d}}\\}$
", "$\\{\\simplify{{inp}}\\in\\mathbb{R}:\\,\\simplify{{inp}>=0}\\}$
", "$\\{\\simplify{{inp}}\\in\\mathbb{R}:\\,\\simplify{{-d}<={inp} <={d}}\\}$
"], "customName": "", "extendBaseMarkingAlgorithm": true, "type": "1_n_2", "prompt": "Given the function \\[\\simplify{{out}({inp})=root(-{inp}^2+{d}{inp},2)},\\] which of the following represents the domain of $\\simplify{{out}}$?
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"}, "variable_groups": [], "variables": {"rat": {"name": "rat", "description": "", "group": "Ungrouped variables", "definition": "if(n=3,'\\\\[\\\\simplify{{out}({inp})=({num[0]}{inp}+{num[1]})/(({inp}-{c[0]})({inp}-{c[1]})*({inp}-{c[2]}))}\\\\]',\nif(n=2,'\\\\[\\\\simplify{{out}({inp})=({num[0]}{inp}+{num[1]})/(({inp}-{c[0]})({inp}-{c[1]}))}\\\\]',\n'\\\\[\\\\simplify{{out}({inp})=({num[0]}{inp}+{num[1]})/(({inp}-{c[0]})({inp}-{c[1]})*({inp}-{c[2]})*({inp}-{c[3]}))}\\\\]'))\n \n \n", "templateType": "anything"}, "n": {"name": "n", "description": "", "group": "Ungrouped variables", "definition": "random(2..4)", "templateType": "anything"}, "a": {"name": "a", "description": "", "group": "Ungrouped variables", "definition": "random(-12..-1)", "templateType": "anything"}, "out": {"name": "out", "description": "", "group": "Ungrouped variables", "definition": "expression(random('f','h','g','p','q','y'))", "templateType": "anything"}, "d": {"name": "d", "description": "", "group": "Ungrouped variables", "definition": "random(1..12)", "templateType": "anything"}, "inp": {"name": "inp", "description": "", "group": "Ungrouped variables", "definition": "expression(random('x','r','s','t','w'))", "templateType": "anything"}, "num": {"name": "num", "description": "", "group": "Ungrouped variables", "definition": "[random(-12..12),random(-12..12 except 0)]", "templateType": "anything"}, "b": {"name": "b", "description": "", "group": "Ungrouped variables", "definition": "a+random(1..12)", "templateType": "anything"}, "c": {"name": "c", "description": "", "group": "Ungrouped variables", "definition": "shuffle(-12..12 except 0)", "templateType": "anything"}, "base": {"name": "base", "description": "base
", "group": "Ungrouped variables", "definition": "random(2..12 except 10)", "templateType": "anything"}}, "extensions": [], "functions": {}, "tags": [], "advice": "a) The domain of $f(x)=\\sqrt{x}$ is the set of all non-negative numbers, i.e. $\\{x\\in\\mathbb{R}:\\,x\\ge0\\}$ or in interval notation, $[0,\\infty)$. This means that the square root of any negative number is not defined.
\n\nb) For a function such as $g(t)=\\sqrt{\\simplify{t-{c[0]}}}$ we require that the square root acts on a non-negative number, that is, $\\simplify{t-{c[0]}>=0}$. Rearranging this inequality for $t$ gives $\\simplify{t>={c[0]}}$. Therefore, $\\text{dom}(g)=\\{t\\in\\mathbb{R}:\\,\\simplify{t>={c[0]}}\\}$.
\n\nc) For a function such as $\\simplify{{out}({inp})}=\\sqrt{\\simplify{{inp}^2-{a+b}{inp}+{a*b}}}$ we require that the square root acts on a non-negative number, that is, $\\simplify{{inp}^2-{a+b}{inp}+{a*b}>=0}$. Notice $\\simplify{{inp}^2-{a+b}{inp}+{a*b}=({inp}-{a})({inp}-{b})}$, which is non-negative when $\\simplify{({inp}-{a})}$ and $\\simplify{({inp}-{b})}$ are both negative, or both non-negative. That is, we require $\\simplify{{inp}<={a}}$ or $\\simplify{{inp}>={b}}$. Therefore, $\\text{dom}(\\simplify{{out}})=\\{\\simplify{{inp}}\\in\\mathbb{R}:\\,\\simplify{{inp}<={a}} \\text{ or } \\simplify{{inp}>={b}}\\}$.
\n\nd) For a function such as $\\simplify{{out}({inp})=root(-{inp}^2+{d}{inp},2)}$ we require that the square root acts on a non-negative number, that is, $\\simplify{-{inp}^2+{d}{inp}>=0}$. Notice $\\simplify{-{inp}^2+{d}{inp}={inp}({d}-{inp})}$, which is non-negative when $\\simplify{{inp}}$ and $\\simplify{({d}-{inp})}$ are both negative, or both non-negative. That is, we require $\\simplify{0<={inp}<={d}}$. Therefore, $\\text{dom}(\\simplify{{out}})=\\{\\simplify{{inp}}\\in\\mathbb{R}:\\,\\simplify{0<={inp}<={d}}\\}$.
\n\n", "name": "Maria's copy of Domain of a square root function", "preamble": {"css": "", "js": ""}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}