Analyzing y as a Function of x

[liuhongwei]

TL;DR

about function

My classmates and I have a big dispute on this question. I think only the first one is right, and they think the third one and the fourth one are right. Please help us to analyze it.

Thank you.

The original title is as follows ：
Which of the following expressions represents y as a function of x?

①2y + x = 3; ②y = x + 2z; ③y = 2; ④y = kx + 1(k is a constant); ⑤y²= x

 

[symbolipoint]

The #4, without any doubt. What further help do you need for this?

#1 could be a function of x, but it is not exactly written that way.

 

[Mark44]

And for the third one, it could be written as y = 0x + 2, making it explicitly a function of x.

 

[mfb]

3 and 4 certainly, and I would say 1 as well (just written in a different way).
2: A matter of definition.
5: No.

 

[member 587159]

You might want to use more descriptive titles.

 

[liuhongwei]

The #4, k=0?

 

[liuhongwei]

3 and 4 certainly, and I would say 1 as well (just written in a different way).
2: A matter of definition.
5: No.

Thank you.
but "y as a function of x ",3 and 4 has none x

 

[liuhongwei]

You might want to use more descriptive titles.

Thanks for your advise

 

[PeroK]

Thank you.
but "y as a function of x ",3 and 4 has none x

What's your definition of "$y$ is a function of $x$"?

 

[jbriggs444]

The way we interpret statements in mathematics differs from the way we interpret spoken statements in ordinary language.

In ordinary language we are constantly called upon to deal with ambiguity and have many unconscious heuristics by which we extract meaning in spite of the ambiguity. One such heuristic is the idea that the speaker is trying to communicate efficiently. For instance, he will prefer to make a stronger statement than a weaker one. He will prefer to make a shorter statement than a longer one.

If someone says "the Cougars won 4 out of their last 5 games", we know that they did not win all five. We also know that there have been at least six games this season and that the lost game was not the first of the last five. We learn as much from what a speaker does not say as from what they do say.

If someone casually says "y is a function of x" we infer that y is not a constant function of x. If it were, the speaker would have said something like "y is a constant", "y depends only on z" or "y does not depend on x".

In mathematical discourse, however, precision is a goal. In a formal context, we avoid making inferences based on what is not said. If someone says "y is a function of x", the technical possibility that it is a constant function must be considered.

 

[PeroK]

Which of the following expressions represents y as a function of x?

①2y + x = 3; ②y = x + 2z; ③y = 2; ④y = kx + 1(k is a constant); ⑤y²= x

To analyse this more precisely: an equation, by itself, does not define a function.
$$2y + x = 3$$
Is only an equation. You may infer a function from it, but that's not explicit in the equation. Whereas,
$$\{(x, y) \in \mathbb{R}^2: 2y + x = 3 \}$$
Does define a function in terms of a subset of $\mathbb{R}^2$ with the required properties. Similarly:
$$\{(x, y) \in \mathbb{R}^2: y = 2 \}$$
also defines a function.

Even $y^2 = x$ may represent a function with the appropriate restriction on $y$.

 

[mfb]

Thank you.
but "y as a function of x ",3 and 4 has none x

Tell me x and I can tell you y. y just happens to be the same for every x, so I don't use your information about x.

 

[mathwonk]

obviously the key point is the definition of what the question means. to me, "y is a function of x", means that knowing the value of x determines the value of y, so to me only 1 and 3 do this. (since we do not know the value of k in 4.) but other interpretations are surely possible.

 

[PeroK]

obviously the key point is the definition of what the question means. to me, "y is a function of x", means that knowing the value of x determines the value of y, so to me only 1 and 3 do this. (since we do not know the value of k in 4.) but other interpretations are surely possible.

So, would you say that
$$f(x) = kx, \ \text{where} \ k \in \mathbb{R}$$
is not function? And, it's only a function if you specify $k$? E.g.
$$f(x) = 2x$$

 

[mathwonk]

I would say that it is a function if k is given. but when written as y = kx, y is only known if both k and x are known. But whether or not the phrase "k is a constant" is sufficient to determine k and make it a function is just to me semantics. This sort of question is not really about mathematics in my opinion. I.e. if you want to specify a function here, you have to give the value of k. Without it, it does not really matter whether or not you are willing to call it a function, you still don't know its values.

I don't mean to claim a definitive expertise about the "right" answer, I just thought I would illustrate that there are many possible opinions, by expressing a new one. I.e. this question is about how hard it is for people to understand each other, it does not really have a right answer, without more information, but unfortunately is the sort of thing that passes for mathematics in some schools.

mathematics is not about using the "right" words for the objects at hand, it is about doing something with those objects.

forgive me, just an old man's 2 cents. everyone is welcome to have a diferent opinion.

edit: I seem to be getting grumpy in the age of the coronavirus.

Ok, I'll grant that y = kx is a function, but not a real valued function, rather it is a function whose values are multiples of k. Of course we all know that to really specify a function, you have to specify the domain, the range, and a rule that determines exactly one element of the range for each choice of an element of the domain. So within set theory, a function is a triple (S,T,F), where S and T are sets, and F is a subset of SxT, such that no two distinct pairs in F have the same first element. In this sense, technically none of those are functions.

 

[zoki85]

② and ⑤ are not

 

[suremarc]

You might have meant no two ordered distinct pairs in F have the same second element. For example, the function whose graph contains the pairs (1, 1) and (-1, 1) would represent a function

Don’t (1, 1) and (-1, 1) have the same second element?

 

[mathwonk]

mark44, as suremarc observes, you might want to reread what you said.

 

[Mark44]

Don’t (1, 1) and (-1, 1) have the same second element?

mark44, as suremarc observes, you might want to reread what you said.

Right - I confused myself. I have deleted my earlier post.

 

[mathwonk]

mark44, you may have made a mistake, but nonetheless, your effort to illustrate it by a simple example was a great way to get at the truth and to communicate clearly. It was this that allowed the clarification of your post. I myself often tend just to rattle on abstractly without examples that would help more than arguments. good instinct!