if $h(x)=x^2+4x+4$ find $h(E)$ if $E=\{ x\in \mathbb R\mid0\leq x\leq1 \}$.

Asked May 27 '21 at 22:39

Active May 28 '21 at 00:21

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If $h(x)=x^2+4x+4$ find $h(E)$ if $E=\{ x \in \mathbb R\mid 0\leq x\leq1 \}.$

It is one of a process on my task on my real analysis lecture, i just stuck at that solve, don't know what to do and have no reference to solve this problem.

Please help me, anyway, thank you very much ^^

edited May 28 '21 at 00:21

Gerry Myerson

179,216

asked May 27 '21 at 22:39

Faqihah Nur Zahirah Muhammad

What have you tried? Maybe try to plot the values $h(0), h(0.1), ..., h(0.9), h(1)$ to get an intuition first. – user594147 May 27 '21 at 22:44
$h$ is continuous and $E$ is connected. So $h(E)$ must be connected. Since $E$ is bounded and closed, then $h$ attains maximum and minimum on $E$. Therefore $h(E)$ is the interval $[\min_{x\in E}(h(x)), \max_{x\in E}(h(x))]$. After this analysis, what you have in your hands is an optimization problem.: Finding maximum and minimum of $h$ on $[0,1]$. Since $h$ is quadratic you could even solve this algebraically, by completing the square. – plop May 27 '21 at 22:55
i have do it just now and got h(0,1) = 4,41 , h(0,2) = 4,84. h(0,3) = 5,29 , h(0,4)=5,76, h(0,5) = 6,25, h(0,6) = 6,76, h (0,7) = 7,29, h(0,8)=7,84, h (0,9)=8,41 , h(1)=9 – Faqihah Nur Zahirah Muhammad May 27 '21 at 23:07
and the max is h(1)=9 and the min is h(0)= 4.. so the h(E) is interval between 4 and 9.. can we know how the function is if we know that two thing? – Faqihah Nur Zahirah Muhammad May 27 '21 at 23:09
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$h(x)$ is $(x+2)^2$. So, you take a number between zero and one, you add two to it, and you square – what do you get? This is not "real analysis"; this is arithmetic. – Gerry Myerson May 28 '21 at 00:21
I'll get the number between 4 and 9.. – Faqihah Nur Zahirah Muhammad May 28 '21 at 03:15
I got it now. Thanks! ^^ – Faqihah Nur Zahirah Muhammad May 28 '21 at 03:29
Anyway, if the question ask for the invers of h(E), what will the answer be? – Faqihah Nur Zahirah Muhammad May 28 '21 at 03:30

if $h(x)=x^2+4x+4$ find $h(E)$ if $E=\{ x\in \mathbb R\mid0\leq x\leq1 \}$.

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