domain of f(x)= 1/(ln(-x^2+4x-3))

{ "query": { "display": "domain $$f\\left(x\\right)=\\frac{1}{\\ln\\left(-x^{2}+4x-3\\right)}$$", "symbolab_question": "FUNCTION#domain f(x)=\\frac{1}{\\ln(-x^{2}+4x-3)}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "domain", "default": "1<x<2\\lor 2<x<3", "interval": "(1,2)\\cup (2,3)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Domain of $$\\frac{1}{\\ln\\left(-x^{2}+4x-3\\right)}\\::{\\quad}1<x<2\\lor\\:2<x<3$$", "steps": [ { "type": "definition", "title": "Domain definition", "text": "The domain of a function is the set of input or argument values for which the function is real and defined" }, { "type": "interim", "title": "Find positive values for logs:$${\\quad}1<x<3$$", "input": "\\frac{1}{\\ln\\left(-x^{2}+4x-3\\right)}", "steps": [ { "type": "step", "primary": "$$\\log_a{f\\left(x\\right)}\\quad\\Rightarrow\\quad\\:f\\left(x\\right)>0$$", "meta": { "general_rule": { "extension": "$$\\log_a{f\\left(x\\right)}$$ has real values only when $$f\\left(x\\right)>0$$" } } }, { "type": "interim", "title": "Solve $$-x^{2}+4x-3>0:{\\quad}1<x<3$$", "input": "-x^{2}+4x-3>0", "steps": [ { "type": "interim", "title": "Factor $$-x^{2}+4x-3:{\\quad}-\\left(x-1\\right)\\left(x-3\\right)$$", "input": "-x^{2}+4x-3", "result": "-\\left(x-1\\right)\\left(x-3\\right)>0", "steps": [ { "type": "step", "primary": "Factor out common term $$-1$$", "result": "=-\\left(x^{2}-4x+3\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "interim", "title": "Factor $$x^{2}-4x+3:{\\quad}\\left(x-1\\right)\\left(x-3\\right)$$", "input": "x^{2}-4x+3", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "x^{2}-4x+3", "steps": [ { "type": "definition", "title": "Definition", "text": "For $$ax^{2}+bx+c\\:$$find $$u,\\:v\\:$$ such that: $$u\\cdot\\:v=a\\cdot\\:c\\:$$and $$u+v=b$$<br/>and group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "secondary": [ "$$a=1,\\:b=-4,\\:c=3$$", "$$u*v=3,\\:u+v=-4$$" ] }, { "type": "interim", "title": "Factors of $$3:{\\quad}1,\\:3$$", "input": "3", "steps": [ { "type": "definition", "title": "Divisors (Factors)", "text": "Factors are numbers we can multiply together to get another number" }, { "type": "interim", "title": "Find the Prime factors of $$3:{\\quad}3$$", "input": "3", "steps": [ { "type": "step", "primary": "$$3$$ is a prime number, therefore no factorization is possible", "result": "=3" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRstjW1xhuS30EMAtGsK3e/Pwt9LEn7QCBUukJKctfSJKeJuqQqR+l8XRZrIWP4OvJvvtJxbo6u/80Nt/bk1au5a/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "primary": "Add 1 ", "result": "1" }, { "type": "step", "primary": "The factors of $$3$$", "result": "1,\\:3" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "Negative factors of $$3:{\\quad}-1,\\:-3$$", "steps": [ { "type": "step", "primary": "Multiply the factors by $$-1$$ to get the negative factors", "result": "-1,\\:-3" } ], "meta": { "interimType": "Negative Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=3,\\:$$check if $$u+v=-4$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=3:\\quad\\:u*v=3,\\:u+v=4\\quad\\Rightarrow\\quad\\:$$False", "secondary": [ "Check $$u=-1,\\:v=-3:\\quad\\:u*v=3,\\:u+v=-4\\quad\\Rightarrow\\quad\\:$$True" ] } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=-1,\\:v=-3" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(x^{2}-x\\right)+\\left(-3x+3\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMwnwvkSFSndwHTUnJVX2i3swsjvX7KVUO/AeCFSId4S337bTjH0JhPfEDoyROh2MmxOR81aIMFSjUznRIvxWbLKFfWxRLPKdJ/qqR++yguicf1RJkcQNMXr+tDbgMzK+EGWWTzraAqG8V/6qnGWqib34sFBI3wQIQWRxYDauqaCe0rVZRNf6poD19Z1jpJbHesg==" } }, { "type": "step", "result": "=\\left(x^{2}-x\\right)+\\left(-3x+3\\right)" }, { "type": "interim", "title": "Factor out $$x\\:$$from $$x^{2}-x:\\quad\\:x\\left(x-1\\right)$$", "input": "x^{2}-x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=xx-x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(x-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "interim", "title": "Factor out $$-3\\:$$from $$-3x+3:\\quad\\:-3\\left(x-1\\right)$$", "input": "-3x+3", "steps": [ { "type": "step", "primary": "Factor out common term $$-3$$", "result": "=-3\\left(x-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YHEwpygaukse3/el9ZtzH5N1pXT08zEQpn0WJ6CFMXD9pNnvGSE4c+RSRHJvBJPw+NpWLk5373GJdyg0yFWbSCGk8vIJisuT2N3pfkW1JpawooBe7OKH4/NA59T9vDuxp6GicmusrFvt0HzBHOUaqrpRe3HwwDDNJPJdMV8UPxU=" } }, { "type": "step", "result": "=x\\left(x-1\\right)-3\\left(x-1\\right)" }, { "type": "step", "primary": "Factor out common term $$x-1$$", "result": "=\\left(x-1\\right)\\left(x-3\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=-\\left(x-1\\right)\\left(x-3\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Multiply both sides by $$-1$$ (reverse the inequality)", "result": "\\left(-\\left(x-1\\right)\\left(x-3\\right)\\right)\\left(-1\\right)<0\\cdot\\:\\left(-1\\right)" }, { "type": "step", "primary": "Simplify", "result": "\\left(x-1\\right)\\left(x-3\\right)<0", "meta": { "solvingClass": "Solver" } }, { "type": "interim", "title": "Identify the intervals", "result": "1<x<3", "steps": [ { "type": "step", "primary": "Find the signs of the factors of $$\\left(x-1\\right)\\left(x-3\\right)$$" }, { "type": "interim", "title": "Find the signs of $$x-1$$", "steps": [ { "type": "interim", "title": "$$x-1=0:{\\quad}x=1$$", "input": "x-1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x-1=0", "result": "x=1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "x-1+1=0+1" }, { "type": "step", "primary": "Simplify", "result": "x=1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "interim", "title": "$$x-1<0:{\\quad}x<1$$", "input": "x-1<0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x-1<0", "result": "x<1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "x-1+1<0+1" }, { "type": "step", "primary": "Simplify", "result": "x<1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": 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$$1$$ to both sides", "result": "x-1+1>0+1" }, { "type": "step", "primary": "Simplify", "result": "x>1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } } ], "meta": { "interimType": "Find Sign 1Eq" } }, { "type": "interim", "title": "Find the signs of $$x-3$$", "steps": [ { "type": "interim", "title": "$$x-3=0:{\\quad}x=3$$", "input": "x-3=0", "steps": [ { "type": "interim", "title": "Move $$3\\:$$to the right side", "input": "x-3=0", "result": "x=3", "steps": [ { "type": "step", "primary": "Add $$3$$ to both sides", "result": "x-3+3=0+3" }, { "type": "step", "primary": "Simplify", "result": "x=3" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": 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to both sides", "result": "x-3+3<0+3" }, { "type": "step", "primary": "Simplify", "result": "x<3" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "$$x-3>0:{\\quad}x>3$$", "input": "x-3>0", "steps": [ { "type": "interim", "title": "Move $$3\\:$$to the right side", "input": "x-3>0", "result": "x>3", "steps": [ { "type": "step", "primary": "Add $$3$$ to both sides", "result": "x-3+3>0+3" }, { "type": "step", "primary": "Simplify", "result": "x>3" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": 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x-3&-&-&-&0&+\\\\\\hline (x-1)(x-3)&+&0&-&0&+\\\\\\hline \\end{array}$$" ] }, { "type": "step", "primary": "Identify the intervals that satisfy the required condition: $$<\\:0$$", "result": "1<x<3" } ], "meta": { "interimType": "Identify The Intervals NoCol 0Eq" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Generic Solve Title 1Eq" } } ], "meta": { "interimType": "Positive Logs 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzmeFdjgvhQTm0mwfI9L6hA/E+YkjtLZw8pc1n2j6Qztb5FbP14/jufBVYISSCumoECXQcz/uASAfLTT8tKitqu3Iduefr1w2DEFJVRhcrO48Z3EgB1hsvNsE46xty+/uw38+Lb7jnCWH9be5n4PC0wj1urFdwtiMS8F2cBsR1R8oVbUezv0YpsjwtNP+V32" } }, { "type": "interim", "title": "Find undefined (singularity) points:$${\\quad}x=2$$", "input": "\\frac{1}{\\ln\\left(-x^{2}+4x-3\\right)}", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{1}{\\ln\\left(-x^{2}+4x-3\\right)}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$\\ln\\left(-x^{2}+4x-3\\right)=0:{\\quad}x=2$$", "input": "\\ln\\left(-x^{2}+4x-3\\right)=0", "steps": [ { "type": "interim", "title": "Apply log rules", "input": "\\ln\\left(-x^{2}+4x-3\\right)=0", "result": "-x^{2}+4x-3=1", "steps": [ { "type": "step", "primary": "Use the logarithmic definition: If $$\\log_a\\left(b\\right)=c\\:$$then $$b=a^c$$", "secondary": [ "$$\\ln\\left(-x^{2}+4x-3\\right)=0\\quad\\:\\Rightarrow\\:\\quad\\:-x^{2}+4x-3=e^{0}$$" ], "result": "-x^{2}+4x-3=e^{0}" }, { "type": "step", "primary": "Simplify", "result": "-x^{2}+4x-3=1" } ], "meta": { "interimType": "Apply Log Rules Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jsEgQ5sg+Qk1yNfOT8EJ5/3vX9ZJIvC0BKfDTH0hGfwFMS0G0HgjUXwc3RQOmn9B9owisEQKkGWeX6altKAVemGR/V5qKo1bMpDx/NFO0ZWFDfPJWQ/4ZJKIJyca2ZNi8LfSxJ+0AgVLpCSnLX0iStslgtbWMspebXSUtXhTKJeGfsvmBMFTMXiD5T+wtHpK" } }, { "type": "interim", "title": "Solve $$-x^{2}+4x-3=1:{\\quad}x=2$$", "input": "-x^{2}+4x-3=1", "result": "x=2", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the left side", "input": "-x^{2}+4x-3=1", "result": "-x^{2}+4x-4=0", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "-x^{2}+4x-3-1=1-1" }, { "type": "step", "primary": "Simplify", "result": "-x^{2}+4x-4=0" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "-x^{2}+4x-4=0", "result": "{x}_{1,\\:2}=\\frac{-4\\pm\\:\\sqrt{4^{2}-4\\left(-1\\right)\\left(-4\\right)}}{2\\left(-1\\right)}", "steps": [ { "type": "definition", "title": "Quadratic Equation Formula:", "text": "For a quadratic equation of the form $$ax^2+bx+c=0$$ the solutions are <br/>$${\\quad}x_{1,\\:2}=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$$" }, { "type": "step", "primary": "For $${\\quad}a=-1,\\:b=4,\\:c=-4$$", "result": "{x}_{1,\\:2}=\\frac{-4\\pm\\:\\sqrt{4^{2}-4\\left(-1\\right)\\left(-4\\right)}}{2\\left(-1\\right)}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$4^{2}-4\\left(-1\\right)\\left(-4\\right)=0$$", "input": "4^{2}-4\\left(-1\\right)\\left(-4\\right)", "result": "{x}_{1,\\:2}=\\frac{-4\\pm\\:\\sqrt{0}}{2\\left(-1\\right)}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=4^{2}-4\\cdot\\:1\\cdot\\:4" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1\\cdot\\:4=16$$", "result": "=4^{2}-16" }, { "type": "step", "primary": "$$4^{2}=16$$", "result": "=16-16" }, { "type": "step", "primary": "Subtract the numbers: $$16-16=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Q7u6Z7DD+L74ggXfh+ZImdMvHyY50dhXPFfrjcmooUgkCDY0JIdcvYqyHrQzk7L1HRHdzS5g2HJFanH3nVRLiMyAayCaqJJzTwjxpVS0tOqwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "x=\\frac{-4}{2\\left(-1\\right)}" }, { "type": "interim", "title": "$$\\frac{-4}{2\\left(-1\\right)}=2$$", "input": "\\frac{-4}{2\\left(-1\\right)}", "result": "x=2", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-4}{-2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{-4}{-2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{4}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{4}{2}=2$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7NtmbHL801OIEJM7rrvtirCRBd7Gm8YO6HN+cGcHYDQqjkVi15I8rBefLi4Iyt2wr5g5ol4Ple/TlcQ5gQvzs1Y81PKJrd/WKXN0NwIfvi0RExyMq+eaFV71EmzDq8qBs" } }, { "type": "step", "primary": "The solution to the quadratic equation is:", "result": "x=2" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The following points are undefined", "result": "x=2" } ], "meta": { "interimType": "Undefined Points 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzmeFdjgvhQTm0mwfI9L6hA/E+YkjtLZw8pc1n2j6QwkeMoBQlGHt0M0LBwI3fCDOkrHIbCbn2nKWek5kAW+M+iyAQ4h/yCDl+R95s7zDj54/wSSw2Uq+t8eJqx4ZKnz+Q1xwoF5inOH16HrjGYrEW5KB+jBxoNJrTrINu1WPPYQB5+CROHR3mx4MIOahKKvwT9/OGp4sbgYfGJwlw8J/rCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "primary": "Combine real regions and undefined points for final function domain", "result": "1<x<2\\lor\\:2<x<3" } ], "meta": { "solvingClass": "Function Domain" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{1}{\\ln(-x^{2}+4x-3)}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }