Question

give a formula for the extended function that is continuous at the indicated point:
f(x) = ( x^2 - 9 ) / ( x + 3 ) , x = -3

Answer 1

Use a piecewise function to specify when x = 3.

Answer 2

hmm. well i know that the answer is y = x - 3, i just have no idea how to get there.

Answer 3

You'd need to add the point at \(x=-3\), using the definition of continuity. So we have:
\[
\lim_{x\to -3}\frac{x^2-9}{x+3}=f(-3)
\]With this, we solve:
\[
\lim_{x\to -3}\frac{x^2-9}{x+3}=\lim_{x\to -3}\frac{(x+3)(x-3)}{x+3}=\lim_{x\to -3}x-3=-6
\]Therefore, at \(x=-3, f(x)=-6\), so we simply add the point to the function to get:
\[
f(x)=\begin{cases}
\frac{x^2-9}{x+3}, x\ne -3\\\\
x=-6, x=-3
\end{cases}
\]

Answer 4

x2−9x+3,x≠−3
x=−6,x=−3

x=−6,x=−3
x=−6,x=−3
x=−6,x=−3