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Calcmathlete
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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.
 2 years ago
 2 years ago
Calcmathlete Group Title
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.
 2 years ago
 2 years ago

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satellite73 Group TitleBest ResponseYou've already chosen the best response.3
for \(f(g(x))\) you must make sure that \(1<g(x)<6\)
 2 years ago

swissgirl Group TitleBest ResponseYou've already chosen the best response.0
I was wrongggg lol
 2 years ago

satellite73 Group TitleBest 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 \[1<x<4\]
 2 years ago

Calcmathlete Group TitleBest 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 \(g(f(x))\)
 2 years ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.3
\(f(g(3))\) is defined because \(1<3<4\)
 2 years ago

Calcmathlete Group TitleBest 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?
 2 years ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.3
now we need to solve \[0<x^2+8x4<4\]
 2 years ago

satellite73 Group TitleBest 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
 2 years ago

satellite73 Group TitleBest 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
 2 years ago

Calcmathlete Group TitleBest 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. \[0 ≤ 11 ≤ 7\]
 2 years ago

satellite73 Group TitleBest 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
 2 years ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.3
i keep putting 4 where there should be a 7, sorry
 2 years ago

Calcmathlete Group TitleBest ResponseYou've already chosen the best response.0
Oh ok. Thank you :) I get it now!
 2 years ago
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