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

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

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

Therefore, ${\textstyle c=1}$ makes ${\textstyle f(x)}$ continuous.