Question

Are f(x) and y the same thing?

This is a philosophical question.

Answer 1

Yes, they are the represent the same thing, except that they sound and look differently.

Answer 2

You wouldn't have a problem with

\[ \large x^2 + (f(x))^2 = 1 \]
?

Answer 3

True... :) Looks abstruse, but it is the same as \(\normalsize\color{blue}{ x^2+y^2=1 }\)

Answer 4

You can factor out of x xD

Answer 5

Doesn't \( f(x) \) mean f is a function of x? But \( x^2 + y^2 = 1\) is not a function. So isn't it not exactly right to put f(x) in place of y?

Answer 6

yes, not in here. I did make an error on that one. But mostly.... I have to admit though that they are different as well.

Answer 7

I meant in general,
\(\normalsize\color{blue}{ f(x)=x.... }\) and \(\normalsize\color{blue}{ y=x.... }\)

Answer 8

I'm sort of musing about why there is such a thing as single-variable calculus. As comes home when you hit implicit differentiation, we are really dealing with at least two variables, although we can stigmatize one by calling dependant in some cases.

Answer 9

Mmm .... looks like IT IS a very philosophical topic, but I just haven't learned about an implicit differentiation yet.

Answer 10

Well, then, look at the students trying to master the slope-intercept equation of a line. Then they are asked "What is the perpendicular?" The answer is always, "the slope of the perpendicular is the negative inverse of the slope." But if they did not solve for y to begin with, they would have the normal (i.e. perpendicular) form.

Answer 11

f (x) is taught to be function notation

Answer 12

x and y is taught as 2D, plotting on Cartesian plane

Answer 13

there is only 1 case when you don't have the slope (if I am understanding the last sentence correctly...) and in this case, the perpendicular LINES are logically easy to find.

Answer 14

I am just speaking from experience and not theory