Science:Math Exam Resources/Courses/MATH100 A/December 2023/Question 02/Solution 1

The two parts of $f(x)$ are continuous, so we only need to make sure that $f(x)$ is continuous at $x=3$. That is $\underset{x\to 3^{-}}{\lim }f(x)=\underset{x\to 3^{+}}{\lim }f(x)=f(3)$. Again, since both parts are continuous this equality will be achieved if the two parts are equal. We set $c\cdot 3+3=c\cdot 3^2-3$, and solve for $c$:

$$\begin{array}{rl}c\cdot 3+3& =c(3{)}^2-3\\ 3c+3& =9c-3\\ 6c& =6\\ c& =1\end{array}$$

Therefore, $c=1$ makes $f(x)$ continuous.