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.
Stacey Warren - Expert brainly.com
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for \(f(g(x))\) you must make sure that \(1
I was wrongggg lol
since \(f\) is only defined for numbers between 1 and 6
\[4>x>-1\] or if you prefer
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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 \(g(f(x))\)
\(f(g(3))\) is defined because \(-1<3<4\)
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?
now we need to solve
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
i think you mean find \(f(3)\) and see if that number is between 0 and 7
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.
\[0 ≤ 11 ≤ 7\]
yes that would make it undefined
if you want to solve
\[0<-x^2+8x-4<4\] you need to solve two separate inequalities,
\[-x^2+8x-4<7\] and take the intersection