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# Domain

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What is the smallest real number in the domain of the function g(x) = sqrt((x - 3)^2 - (x - 18)^2)?

Apr 9, 2021

#1
+321
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21/2

Apr 9, 2021
#2
+496
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If we want a real number, then ${(x-3)^2-(x-18)^2}$ has to be greater than or equal to $0$, or $(x-18)^2$ has to be less than or equal to $(x-3)^2$

when $(x-3)^2=(x-18)^2$, x has the smallest possible value with real solutions. So we have:

$(x-3)^2=(x-18)^2=x^2+9-6x=x^2+324-36x$

$30x=315$

$x=10.5$

so if x is any less than $\boxed{10.5}$, then the equation $\sqrt{(x - 3)^2 - (x - 18)^2}$ would have no real solutions.

Apr 9, 2021
edited by SparklingWater2  Apr 9, 2021