Science:Math Exam Resources/Courses/MATH100/December 2016/Question 09 (a) (i)

MATH100 December 2016

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[hide] Question 09 (a) (i)
Let $f(x)={\frac {\sqrt {x^{2}-8}}{x-4}}$. (i) What is the domain of $f(x)$?

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

[show] Hint 1
What's the domain of the square root function?

[show] Hint 2
A quotient of two functions is defined where both the numerator and denominator are defined and the denominator is non-vanishing.

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[show] Solution

First consider the numerator. Since the square root function is defined only for non-negative real numbers, the numerator is defined only when $x^{2}-8\geq 0$, i.e., when $x\geq {\sqrt {8}}=2{\sqrt {2}}$ or $x\leq -2{\sqrt {2}}$, that is to say, when $x\in (-\infty ,-2{\sqrt {2}}]\cup [2{\sqrt {2}},\infty ).$ On the other hand, the denominator is defined everywhere on the real line. But it vanishes at $x=4$. Therefore, the domain of $f$ is $\color {blue}\left((-\infty ,-2{\sqrt {2}}]\cup [2{\sqrt {2}},\infty )\right)\setminus \{4\}$.