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

MATH100 A December 2023

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Question 02
Find the value of the constant ${\textstyle c}$ that makes ${\textstyle f(x)}$ continuous, where $$f(x)=\left\lbrace {\begin{array}{ll}cx+3&{\text{if }}x\leq 3\\cx^{2}-3&{\text{if }}x>3\end{array}}\right.$$

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
This is a piecewise function, what needs to happen at the boundary of the pieces for a piecewise function to be continuous?

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. 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.