Continuity Question with Piecewise Defined Function and Absolute Values

**joatmon** · February 5th 2011, 07:54 PM

Determine the values of b and c so that the following function is continuous:

$f(x) = \left\{\begin{array}{cc}x+1,&\mbox{ if }<br /> 1< {x}< 3\\x^2 + bx + c, & \mbox{ if } \lvert x+2 \rvert \geq 1\end{array}\right$

Restating this function to remove the absolute values, I get:

$f(x) = \left\{\begin{array}{cc}x^2 + bx + c,&\mbox{ if }<br /> {x}\leq -3\\ x^2 + bx + c, & \mbox{ if } x \geq -1\\x + 1 , &\mbox{ if } 1 < x < 3 \end{array}\right$

But this makes absolutely no sense to me since I can see no way that this function can be continuous defined like I have done it. I must be screwing up the absolute value inequality.

Can somebody help?

Thanks.

**Prove It** · February 5th 2011, 08:27 PM

Are you sure that it's not $\displaystyle x^2 + bx + c$ if $\displaystyle |x - 2| \geq 1$ ?

Because then $\displaystyle x - 2 \leq -1$ or $\displaystyle x - 2 \geq 1$

$\displaystyle x \leq 1$ or $\displaystyle x \geq 3$ .

This would at least ensure that your function is at least DEFINED for all values over your given domain...

**joatmon** · February 5th 2011, 10:54 PM

It says x + 2, but that has to be a typo. I'm going to treat it as x - 2, since it's the only way that I know how to solve this problem.

Thanks.

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