Answer to Question #263373 in Calculus for Sabelo Xulu

Question #263373

The functions f and g are defined by f(x)=1/|1-3x| and g(x)=log1/3(1 /3x-2)-log3(x) respectively

1.1. Write down the sets Df(the domain of f) and Dg(the domain of g)

1.2. Solve the inequality f(x) greater than 2 for x"\\in" Df

1.3. Solve the inequality g(x) greater or equals to 0 for x"\\in" Dg

Hint: use the change of base formula


1
Expert's answer
2021-11-10T10:15:22-0500

1.1 Domain of the function is the set of all values that argument might take. So

"f(x)={\\frac 1 {|1-3x|}}" , then "|1 - 3x| \\not=0\\implies x \\not ={\\frac 1 3}"

"D{\\scriptscriptstyle f}:x \\isin R \/" {"{\\frac 1 3}"}

"g(x) = \\log _{{\\frac 1 3}} {{\\frac 1 {3x-2}}}-\\log _{3} x" , then "3x-2>0" and "x>0"

"D{\\scriptscriptstyle g}:x \\isin ({\\frac 2 3};+\\infty)"


1.2 "{\\frac 1 {|1-3x|}}>2\\implies |1-3x|<0.5\\implies (1-3x < 0.5)\\land (1-3x>-0.5)\\implies x \\in ({\\frac 1 6}; {\\frac 1 2})"

Considering the domain, the answer is "x \\in ({\\frac 1 6}; {\\frac 1 2})" / {"{\\frac 1 3}"}


1.3 "\\log _{{\\frac 1 3}} {{\\frac 1 {3x-2}}}-\\log _{3} x\u22650\\implies -\\log _{3} {{\\frac 1 {3x-2}}}-\\log _{3} x\u22650\\implies-(\\log _{3} {{\\frac 1 {3x-2}}}+\\log _{3} x)\u22650"

"-(\\log _{3} {{\\frac 1 {3x-2}}}+\\log _{3} x)\u22650\\implies \\log _{3} {{\\frac 1 {3x-2}}}+\\log _{3} x\u22640\\implies \\log _{3} {{\\frac x {3x-2}}} \u22640\\implies {\\frac x {3x-2}}\u22641"

"{\\frac {x-3x+2} {3x-2}}\u22640 \\implies {\\frac {-2x+2} {3x-2}}\u22640\\implies ((-2x+2\u22640)\\land (3x-2>0))\\lor((-2x+2 \u22650)\\land( 3x-2 < 0))\\implies ((x\u22651)\\land (x>{\\frac 2 3}))\\lor ((x\u22641)\\land (x<{\\frac 2 3}))\\implies (x\u22651)\\lor (x<{\\frac 2 3})"

Considering the domain, the answer is "x\\isin [1;+\\infty) \u222a (-\\infty;{\\frac 2 3})"


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
APPROVED BY CLIENTS