Find the Unknown in Piecewise Function | Calculus

Question:

For what value of $k$,$$f(x)=\{\begin{array}{ll}4x^2-kx-10& 0\le x<2\\ 1-6x& 2\le x\le 3\end{array}$$ $\text{is continuous}\mathrm{\forall }x\in [0,3]$.

Answer:

If the function is continuous, the limit at the break points must evaluate to the same value. Therefore, the following condition must be met, $$\underset{x\to 2^{-}}{lim}4x^2-kx-10=\underset{x\to 2^{+}}{lim}1-6x$$ $$\Rightarrow 4(2{)}^2-k(2)-10=1-6(2)$$ $$\Rightarrow 6-2k=-11$$ $$\Rightarrow 17=2k$$ $$\Rightarrow k=\frac{17}{2}$$ Secondly, since the piecewise function consists of polynomials, we know that the polynomials are continuous for their respective domains. Hence, the value of k must be $k=\frac{17}{2}$ if the function $\text{is continuous}\mathrm{\forall }x\in [0,3]$.