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# domain and range

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What is the smallest real number x in the domain of the function g(x) = sqrt((x + 3)^2 - (x - 8)^2)?

Feb 12, 2021

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We notice the value under the square root must be positive to be in the domain. So we have \$(x+3)^2-(x-8)^2\geq0\$. We see that after simplifying, \$22x-55\geq0\$, where the minimum of \$x\$ is then \$2.5\$. Clearly, the smallest \$x\$ in the domain is \$2.5\$.

Feb 12, 2021