A major goal of this course is for you to build a solid foundation of functions. This will not happen overnight, and it won’t happen without failure. One of the most important things you can learn from this course is how to learn from your failure. The exams are designed with this in mind.

Read over an example question/answer from a key provided below.

First, let’s solve it correctly. To find the domain of a radical function, it is either:

• If the root is even, set the radicand (underneath the radical) $\geq 0$ and solve (as we can’t take the even root of a negative number in the Real numbers).
• If the root is odd, the domain is all Real numbers (as we can take the odd root of a negative number).

Thus, the domain is all Real numbers and the solution is E.

The options A, B, C, D, E are all exactly as they appear on the exam. The text underneath each option gives you an idea of why a student might have chosen this option. Let’s look at each one more closely:

• It is common for students to begin by setting the radicand $\geq 0$ before considering whether the root is even or odd. If a student did this, they would be solving the inequality $-4x-3 \geq 0$. A common mistake at this point could be to do this in your head and switch the order of the solution fraction to be $x \leq -\frac {4}{3}$ rather than $x \leq -\frac {3}{4}$ if they had written out the steps. So a student who (1) does not check if the root is even or odd AND (2) makes the simple mistake of flipping the fraction solution would choose this option.
• Let’s say a student did everything that student A did in the previous option, but forgot to flip the inequality when dividing by a negative number. They would get the inequality $x \geq -\frac {4}{3}$ and thus choose this option. Notice this option contains a new, very specific common mistake.
• We can also think of a student who’s only mistake was treating the domain of all radical functions as setting the radicand $\geq 0$ and solving.
• Finally, we can think of a student who did not check the root of the radical and did not flip the inequality when dividing by a negative number.

So, what does this mean?

It is difficult to say definitively with just this question. A student who chose option C may have been rushing, may have been flustered, or may simply not know that radical functions have different domains based on the power of the root. But if you are reviewing your own results, soon after taking the exam, you would know which of this was the case.

And if you chose option A or B? You had a couple of simple arithmetic errors, but also had a foundational misconception about domains of radical functions.

You can now see how the multiple-choice questions can grade and give insights like a free-response question.