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Given a randomised log function select the possible ways of writing the domain of the function.

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Given the real functions below, you should be able to determine their domains.

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Give an example of a real number that is not in the domain of the function

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\$f(x)=\\log(x).\$

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[[0]] $\\notin \\text{dom}(f)$

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$t>\\var{c[0]}$

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$t\\ge \\var{c[0]}$

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$t<\\var{c[0]}$

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$t\\le \\var{c[0]}$

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$t>0$

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$t\\ne0$

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Given the function \$g(t)=\\log_{\\var{base}}(\\simplify{t-{c[0]}}),\$ what do we require of $t$ so that $g(t)$ is defined?

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$[\\var{c[3]},\\infty)$

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$(0,\\infty)$

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$\\{s\\in\\mathbb{R}:\\, s\\ge 10\\}$

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$\\{s\\in\\mathbb{R}:\\, s> \\simplify[fractionNumbers]{{-c[2]/c[1]}}\\}$ $\\{s\\in\\mathbb{R}:\\, s< \\simplify[fractionNumbers]{{-c[2]/c[1]}}\\}$

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$\\{s\\in\\mathbb{R}:\\, s< \\var{c[3]}\\}$

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$\\{s\\in\\mathbb{R}:\\, s< \\simplify[fractionNumbers]{{-c[2]/c[1]}}\\}$ $\\{s\\in\\mathbb{R}:\\, s> \\simplify[fractionNumbers]{{-c[2]/c[1]}}\\}$

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Given the function \$h(s)=\\simplify{{c[0]}log({c[1]}s+{c[2]})+{c[3]}},\$ which of the following represents the domain of $h$?

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$\\{\\simplify{{inp}}\\in\\mathbb{R}:\\,\\simplify{{inp}<{a}} \\text{ or } \\simplify{{inp}>{b}}\\}$

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$\\{\\simplify{{inp}}\\in\\mathbb{R}:\\,\\simplify{{a}<{inp} <{b}}\\}$

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$\\{\\simplify{{inp}}\\in\\mathbb{R}:\\,\\simplify{{inp}>{a*b}}\\}$

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$\\{\\simplify{{inp}}\\in\\mathbb{R}:\\,\\simplify{{inp}>0}\\}$

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$\\{\\simplify{{inp}}\\in\\mathbb{R}:\\,\\simplify{{min([a+b,a*b])}<{inp} <{max([a+b,a*b])}}\\}$

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Given the function \$\\simplify{{out}({inp})=ln({inp}^2-{a+b}{inp}+{a*b})},\$ which of the following represents the domain of $\\simplify{{out}}$?

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$\\{\\simplify{{inp}}\\in\\mathbb{R}:\\,\\simplify{{inp}<0} \\text{ or } \\simplify{{inp}>{d}}\\}$

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$\\{\\simplify{{inp}}\\in\\mathbb{R}:\\,\\simplify{0<{inp} <{d}}\\}$

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$\\{\\simplify{{inp}}\\in\\mathbb{R}:\\,\\simplify{{inp}>{d}}\\}$

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$\\{\\simplify{{inp}}\\in\\mathbb{R}:\\,\\simplify{{inp}>0}\\}$

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$\\{\\simplify{{inp}}\\in\\mathbb{R}:\\,\\simplify{{-d}<{inp} <{d}}\\}$

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Given the function \$\\simplify{{out}({inp})=ln(-{inp}^2+{d}{inp})},\$ which of the following represents the domain of $\\simplify{{out}}$?

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base

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a) The domain of $f(x)=\\log(x)$ is the set of all positive numbers, i.e. $\\{x\\in\\mathbb{R}:\\,x>0\\}$ or in interval notation, $(0,\\infty)$. This means that the log of zero, or any negative number is not defined. This is true regardless of the base of the logarithm.

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b) For a function such as $g(t)=\\log_{\\var{base}}(\\simplify{t-{c[0]}})$ we require that log acts on a positive number, that is, $\\simplify{t-{c[0]}>0}$. Rearranging this inequality for $t$ gives $\\simplify{t>{c[0]}}$. Therefore, $\\text{dom}(g)=\\{t\\in\\mathbb{R}:\\,\\simplify{t>{c[0]}}\\}$.

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c) For a function such as $h(s)=\\simplify{{c[0]}log({c[1]}s+{c[2]})+{c[3]}}$ we require that log acts on a positive number, that is, $\\simplify{{c[1]}s+{c[2]}>0}$. Rearranging this inequality for $s$ gives $s> \\simplify[fractionNumbers]{{-c[2]/c[1]}}$ $s< \\simplify[fractionNumbers]{{-c[2]/c[1]}}$. Therefore, $\\text{dom}(h)=\\{s\\in\\mathbb{R}:\\, s> \\simplify[fractionNumbers]{{-c[2]/c[1]}}\\}$ $\\text{dom}(h)=\\{s\\in\\mathbb{R}:\\, s< \\simplify[fractionNumbers]{{-c[2]/c[1]}}\\}$.

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d) For a function such as $\\simplify{{out}({inp})=ln({inp}^2-{a+b}{inp}+{a*b})}$ we require that log acts on a positive number, that is, $\\simplify{{inp}^2-{a+b}{inp}+{a*b}>0}$. Notice $\\simplify{{inp}^2-{a+b}{inp}+{a*b}=({inp}-{a})({inp}-{b})}$ is positive when $\\simplify{({inp}-{a})}$ and $\\simplify{({inp}-{b})}$ are both negative, or both positive. That is, we require $\\simplify{{inp}<{a}}$ or $\\simplify{{inp}>{b}}$. Therefore, $\\text{dom}(\\simplify{{out}})=\\{\\simplify{{inp}}\\in\\mathbb{R}:\\,\\simplify{{inp}<{a}} \\text{ or } \\simplify{{inp}>{b}}\\}$.

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e) Note $\\ln$ is just the notation for $\\log_e$. For a function such as $\\simplify{{out}({inp})=ln(-{inp}^2+{d}{inp})}$ we require that log acts on a positive number, that is, $\\simplify{-{inp}^2+{d}{inp}>0}$. Notice $\\simplify{-{inp}^2+{d}{inp}={inp}({d}-{inp})}$ is positive when $\\simplify{{inp}}$ and $\\simplify{({d}-{inp})}$ are both negative, or both positive. That is, we require $\\simplify{0<{inp}<{d}}$. Therefore, $\\text{dom}(\\simplify{{out}})=\\{\\simplify{{inp}}\\in\\mathbb{R}:\\,\\simplify{0<{inp}<{d}}\\}$.

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