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Let \(f\) and g be defined by\[f(x) = x^2 + 8x  4~~~~~1 ≤ x ≤ 6\]\[g(x) = 5  x~~~~~0 ≤ x ≤ 7\]How would I go about with finding the domains of \(f(g(x))\) and \(g(f(x))\)? Also, how would I prove that \(f(g(3))\) is defined and \(g(f(3))\) is undefined? More than likely, it has to do with the domain, but I'm not fully sure with how to do it.
 one year ago
 one year ago
Let \(f\) and g be defined by\[f(x) = x^2 + 8x  4~~~~~1 ≤ x ≤ 6\]\[g(x) = 5  x~~~~~0 ≤ x ≤ 7\]How would I go about with finding the domains of \(f(g(x))\) and \(g(f(x))\)? Also, how would I prove that \(f(g(3))\) is defined and \(g(f(3))\) is undefined? More than likely, it has to do with the domain, but I'm not fully sure with how to do it.
 one year ago
 one year ago

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satellite73Best ResponseYou've already chosen the best response.3
for \(f(g(x))\) you must make sure that \(1<g(x)<6\)
 one year ago

satellite73Best ResponseYou've already chosen the best response.3
since \(f\) is only defined for numbers between 1 and 6 solve via \[1<5x<6\] \[4<x<1\] \[4>x>1\] or if you prefer \[1<x<4\]
 one year ago

CalcmathleteBest ResponseYou've already chosen the best response.0
Yes. I got for that: \[Domain~of~f(g(x) = 1 ≤ x ≤ 4\]I seem to get irrational answers for the other part though as in \(g(f(x))\)
 one year ago

satellite73Best ResponseYou've already chosen the best response.3
\(f(g(3))\) is defined because \(1<3<4\)
 one year ago

CalcmathleteBest ResponseYou've already chosen the best response.0
Oh. Could I just plug in 3 here: \[g(f(x)) = 0 ≤ x^2 + 8x  4 ≤ 7\]and see if it is true or not then?
 one year ago

satellite73Best ResponseYou've already chosen the best response.3
now we need to solve \[0<x^2+8x4<4\]
 one year ago

satellite73Best ResponseYou've already chosen the best response.3
well that would answer the question for sure, but it would not find the domain of \(g\circ f\) it would only show that 3 is not in it
 one year ago

satellite73Best ResponseYou've already chosen the best response.3
i think you mean find \(f(3)\) and see if that number is between 0 and 7
 one year ago

CalcmathleteBest ResponseYou've already chosen the best response.0
Well I now know how to find the domain of it if I wanted to, but this is a no calculator question, so it'd be hard to gauge the domain. I think I get it now because this is not true, it is undefined. \[0 ≤ 11 ≤ 7\]
 one year ago

satellite73Best ResponseYou've already chosen the best response.3
yes that would make it undefined if you want to solve \[0<x^2+8x4<4\] you need to solve two separate inequalities, \[0<x^2+8x4\]and \[x^2+8x4<7\] and take the intersection
 one year ago

satellite73Best ResponseYou've already chosen the best response.3
i keep putting 4 where there should be a 7, sorry
 one year ago

CalcmathleteBest ResponseYou've already chosen the best response.0
Oh ok. Thank you :) I get it now!
 one year ago
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