Question

5. Given the function

Answer 1

The function

$f(x)=\begin{cases}cx^2&\text{for }x\le2\\cx-3&\text{for }x>2\end{cases}$

is piecewise continuous, since both cx ² and cx - 3 are polynomials. f(x) itself is continuous if both pieces meet at the same defined point. In other words, the limits of f(x) as x → 2 from either side have the same value of f (2) = c•2² = 4c.

We have

$\displaystyle\lim_{x\to2^-}f(x)=\lim_{x\to2}cx^2=4c$

$\displaystyle\lim_{x\to2^+}f(x)=\lim_{x\to2}(cx-3)=2c-3$

so in order for f to be continuous, we need

4c = 2c - 3   →   2c = -3   →   c = -3/2