Finding the Domain II

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Last time we looked at finding the domain of quotient functions, and we saw that we had to make sure that the denominator was never zero. This isn’t the only thing that can go wrong, though, so let’s take a look at a few more examples.

Check out the function f(x)=\sqrt{x}. What is its domain? It’s not all real numbers, since \sqrt{-1} is not a real number. (If you happen to recognize this as a perfectly nice complex number, pat yourself on the back, but remember that -1 still isn’t in the domain of f. In college calculus we usually just consider real numbers. )

So we can’t put in -1, or any other negative number, for x in f. Positive numbers work, though, and so does zero. Therefore the domain of f is all real numbers greater than or equal to zero.

Here’s another example of a function whose domain is restricted: g(x)=\ln (x-3). The natural logarithm is defined only for positive values of its argument. That means that we need x-3 > 0, or in other words x > 3. The domain of g is all real numbers greater than three.

Here are some more questions on finding the domain:

  1. What is the domain of h_1(x)=\sqrt{x^2+4}? What about h_2(x)=\sqrt{x^2-4}?
  2. What is the domain of u(x)=\ln (2x-9)?
  3. What is the domain of v(x)=\sqrt{\sin x}?