Test 1 Section 4 #36

You’re in the calculator section, so the *fastest* way to get this is by graphing. Just type the function in carefully, and graph it:

Those vertical asymptotes around *x* = 3 tell you that 3 is probably the answer. Want to make sure? Look at your Table:

Yep, the function is undefined at *x *= 3.

Whenever I post a calculator-based solution, though, people always ask me how to solve *without* a calculator. Of course, there’s a way, but here is a good place for me to recommend to you that if you don’t have one of your own, you find a calculator you can use on test day (maybe you can borrow one from school) and take the time *now* to get used to using it. Anyway, the math:

What they’re trying to show you there with the (*x* – 5)s is that you might want to consider them a whole value and simplify this by substituting for (*x* – 5). Let’s say (*x* – 5) = *p* for a moment. Then we can say:

Of course, that’s a binomial square, so we can say:

Now reintroduce (*x* – 5) for *p*:

And that is why the function is undefined at *x* = 3: the denominator equals zero when *x* = 3.

Note that you can also simplify to if you expand the denominator rather than doing the substitution I just did. That’s not too much more labor intensive and is arguably more straightforward. That solution looks like this: