For what value of c makes the function continuous? [closed]

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My answer here is d only because I graphed the piecewise function and tried to plug in the choices. None of the given values make the function continuous for all x. Is my answer right? Is there a way to solve this without graphing?

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1 Answer

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Observe the definitions of the pieces in which the function is defined.

$2-sinx$ as well as $cx\sqrt{x^2+2}+c$ are continuous in their domains. So we need to check the continuity only at $x=1$. So, by equating both the sides of the function at $x=1$, we have:

$$2-sin1=c\sqrt3+c$$

$$c\approx0.424$$

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