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If $f(x)$ is a function whose domain is $[-8,8]$, and $g(x)=f\left((x^2 - 2)/(x + 1)\right)$, then the domain of $g(x)$ is an interval of what width?

 Apr 15, 2024
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We need to analyze when the expression inside the function f in g(x) becomes undefined. This happens when the denominator, (x+1), is zero. So, x=−1.

However, this restriction only applies if the numerator, (x2−2), is not also zero at x=−1. Since (−1)2−2=1=0, the function f((x2−2)/(x+1)) is defined even when x=−1.

Therefore, the only restriction on the domain of g(x) is that x=−1. This means g(x) is defined for all real numbers in the interval except for this single point.

The width of an interval that excludes a single point is 0​.

 Apr 15, 2024

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