Section 5.4 Closing Ideas
The use of variables to represent other variable expressions is an extremely important skill in mathematics. In your college math course, you will likely see that function notation can lead you to make the exact same type of substitutions. Here is a brief preview of that.
Suppose we define the function \(f(x) = x^2 - 2\text{.}\) To evaluate the function means to substitute for the variable and simplify the result. Here are some examples.
\(\displaystyle f(y) = y^2 - 2\)
\(\displaystyle f(x - 3) = (x - 3)^2 - 2\)
But students start to struggle with this concept as soon as we start substituting objects other than numbers:
\(\displaystyle f(y) = y^2 - 2\)
\(\displaystyle f(x - 3) = (x - 3)^2 - 2\)
(In the second case, you might be asked to simplify the expression.)
The actual concept involved is exactly what we've done in this section. We've simply replaced the variable of the function with a variable expression. But students who lack a clear understanding of this concept end up with all sorts of wrong answers. Here are some examples:
\(\displaystyle f(x - 3) = (x^2 - 3) - 2\)
\(\displaystyle f(x - 3) = x - 3^2 - 2\)
\(\displaystyle f(x - 3) = (x^2 + 2) - 3\)
These sorts of erroneous manipulations stem from a much deeper place than most students realize. Some students don't even want to try to think through the problem, and so they just guess and hope for the best. And there's some human psychology to this. They feel so accustomed to being wrong in math classes that they've simply given up trying to be right. Their mentality has shifted to "I'm just going to write down something and wait for someone to tell me what the right thing is." And so they just scribble down symbols and hope for the best. As we keep making our way through the different topics, remember that mathematical thinking includes having a clear sense of what you're doing and why you're doing it. Everything is supposed to be logical, and everything is supposed to make sense.
Most of the errors above stem from students simply not taking the time to think carefully. In fact, many students can find their errors after they start to try to explain what they're doing and why they're doing it. In other words, most students know enough to fix their own errors, if they would just take the time to think more carefully about it.
Success in mathematics is attainable through the consistent effort of careful and thoughtful practice. It's not about being fast or even being smart. Math is a learned skill like every other learned skill, which means that you can learn it by putting in the time and the effort. So do not sell yourself short by refusing to try to learn. You're going to make mistakes, and that's okay. Just keep trying to do a little bit better every time.