Mistakes regarding functions

Concerning confusion between single variable and multivariable functions

• I have no idea if this mistake happens, but it's a possibility nonetheless. Suppose that we have [math]\displaystyle{ f(x,y) = x + y }[/math]. It's possible that one sees [math]\displaystyle{ g(x) = x }[/math] and [math]\displaystyle{ h(y) = y }[/math], then [math]\displaystyle{ f(x,y) = g(x) + h(y) }[/math]. The numerical results are the same for both the two variable function and the sum of two single variable functions. However, we can't just split a multivariable function like that! By doing so we are, inadvertently, considering that we have different functions governing different processes. Each function having its own particular domain. One practical case is to consider the square root or trigonometric functions. For some pairs of numbers it may hold true, but we can quickly see that a sum of roots or sum of sines won't be the same as the sine of the sum or the root of the sum.

• [math]\displaystyle{ f(x,y) = x^2 }[/math] is this a single variable function or a multivariable one? It's multivariable. The fact that the equation doesn't have [math]\displaystyle{ y }[/math] is not the same as to say that "the second variable doesn't exist". It's there. Where? It's a constant, the null (zero) constant.

Concerning the piecewise function

• Is there a difference between having a piecewise function with multiple cases and multiple functions with multiple intervals? For example: define a parabola for [math]\displaystyle{ x \lt 2 }[/math], constant function for [math]\displaystyle{ 2 \leq x \lt -2 }[/math] and another parabola for [math]\displaystyle{ x \geq 2 \ ? }[/math] Graphically it won't matter. But conceptually, one piecewise function versus three functions may lead to the interpretation of three different processes rather than one single process.

Concerning properties

• [math]\displaystyle{ f(x) = 3 }[/math] is a constant function and the number 3 is odd. But this function is even. It's a bit misleading.

Concerning the zero and the empty set

• I don't know if this confusion happens, but people may think that if a function is undefined at a point, it's the same as to say that the function is equal to zero there. Not quite the same thing. Zero is a number and if a set contains the zero alone, it's not an empty set because it contains something.

Concerning level curves

• When we do [math]\displaystyle{ f(x,y) = c }[/math], a confusion that can happen is to think that [math]\displaystyle{ c = x = y }[/math]. We aren't making the function's variables constants, what we are regarding as constant is the image. What we are plotting is every pair [math]\displaystyle{ (x,y) }[/math] for which the function has a certain constant value. In the same way more than one [math]\displaystyle{ x }[/math] can map to one, and only one, [math]\displaystyle{ f(x) }[/math]. Multiple values for the dependent variables can result in the same constant with multivariable functions.

Concerning the absolute value

• This mistake arises from the fact that when we first learn about modulus, we don't know what a function is yet. Many teachers and books say that "modulus erases the minus sign" of a number. It's better to say that the modulus is a function, because modulus is really a piecewise function. So we have that [math]\displaystyle{ |2| = |-2| }[/math]. That definition "modulus is an operation that removes the minus sign" can lead us to think that "minus two is equal to two". When the correct way to read it is "The modulus of minus two is equal to the modulus of two". A very subtle difference! Nobody sees the expression [math]\displaystyle{ -2 = 2 }[/math] as true because it isn't. However, the consequence of misreading the modulus is wrongly plotting graphs of functions. Suppose that we have this function [math]\displaystyle{ f(x) = |x| }[/math]. When we think that "minus two is equal to two" we can, inadvertently, plot 2 and -2 at the same position. What is equal there is [math]\displaystyle{ f(-2) = f(2) \ !!! }[/math]. It's a quirk that happened to myself more than once.
  
  I've never liked the definition that modulus erases the sign of a number. We all learn that numbers are either positive, negative or zero. What is the "absolute number"?? Something that is neither positive nor negative? This is the confusion that comes along absolute value when we are unaware of functions.

Concerning reading and graphs

• Crescent, decrescent and constant functions. First, for languages that are written right to left this may be a source of confusion in comparison to languages that are left to right. The other is about reading the graph itself. It my happen that some people associate in their minds "positiveness" with a function being crescent and "negativeness" with being decrescent. While "null" would mean constant. Yet another source of confusion is to think that concavity up or concavity down are, respectively, synonyms for crescent and decrescent. I'm only speculating, but some people may even make the association between a ramp and acceleration and confuse crescent with decrescent and vice-versa because of that.
  
  I think this misconception comes from the way teachers at school teach functions. A function is crescent in a certain interval not because all values of [math]\displaystyle{ f(x) }[/math] in that interval are positive, but because the rate of change is positive. A function is decrescent in a certain interval because its rate of change is negative, not because [math]\displaystyle{ f(x) }[/math] lies within the negative part of the vertical axis. Lastly, being constant means a zero rate of change, not literally [math]\displaystyle{ f(x) = 0 }[/math].

• Suppose that the graph of a function displays [math]\displaystyle{ f(b) \gt f(a) }[/math] with [math]\displaystyle{ b \gt a }[/math]. That does not mean that the function is always crescent from [math]\displaystyle{ a }[/math] to [math]\displaystyle{ b }[/math]. The graph may have points where changes from crescent to decrescent and vice-versa. For the same reason [math]\displaystyle{ f(b) \lt f(a) }[/math] doesn't guarantee that the function is always decrescent in that interval.

• There is a common mistake that really demonstrates that people don't understand the concept of a function at all. Suppose that a problem describes something such as the population's growth, the variation in temperature in respect to time or the brightness of a light in respect to electrical power. The problem gives the graph, which excuses us from having to gather data and then finding the function. Now the question is something such as: beginning at 2 hours, after 1 hour, what is the temperature going to be? For a brightness of x units, what is the electrical power? What's the expected population at year x? Not knowing to answer these questions while the graph is given means that people really don't understand functions at all. Now the curious fact is that maybe the same people are able to do the calculations and solve the equations.

• Reading the graph of a single variable function is pretty easy. Nonetheless, from time to time, some people confuse [math]\displaystyle{ x }[/math] with [math]\displaystyle{ f(x) }[/math]. This is specially problematic with physics. Suppose we have a function that depicts Speed x Time. It's not uncommon for one to read the graph mixing up time with speed, pointing out the wrong values when asked about time or speed. When we learn about derivatives and integrals it becomes even worse! It's not uncommon for people to confuse the graphs of each one because then we have not just one function, but three and three graphs.

• The other common mistake is to confuse position or trajectory with velocity or acceleration. Suppose that the function is a straight line that begins at some time and v(t) = -10 m/s. At some later time it's v(t) = 10 m/s. First, who said that the trajectory is a straight line? The graph is Velocity x Time. Second, velocity is a vector, the sign indicates the direction, not the magnitude. Third, by integrating v(t) we obtain x(t) and the graph should be parabola. However, who said that x(t) is a graph of a trajectory? The motion itself is not the parabola! It's the measure, the distance from a certain origin, that is behaving as a parabola.

• Trajectories x plane curves x graph of a function. It's not uncommon for people to confuse equations of lines or planes with functions of two or three variables.
  
  Suppose a person is running in circles on the Cartesian plane. That circle is described by an equation of a circle, like this: [math]\displaystyle{ 4 = x^2 + y^2 }[/math]. It's a circle with radius = 2 with the center at the origin (0,0). Later, when we learn about functions of two variables, that same equation is called a level curve. This [math]\displaystyle{ f(x,y) = x^2 + y^2 - 4 }[/math] is a function of two variables, not the equation of a circle!
  
  Careful with [math]\displaystyle{ y = x }[/math] and [math]\displaystyle{ y = f(x) }[/math]. The former can mean two things: that two numbers are equal to each other or the equation of a line, in which the coordinates of all its points obey to that equality. Now the latter is a way to say this "the vertical coordinate of each and every point of the function's graph is given by calculating the function at that [math]\displaystyle{ x }[/math]".
  
  To clarify the confusion between the XY Cartesian plane and the "X x f(X)" plane I'd say this: the XY plane has two independent axis where we can freely choose any number we want from one axis and from the other to find positions, points. That's why we can draw circles on it. Y does not depend on X and vice-versa. On the other hand, to plot functions, the vertical axis is always dependant on the values of the horizontal axis. That's why functions can only increase, decrease or remain constant. We either go forwards or backwards. If we "move" X to the right, f(x) goes along with it. The same for "moving" to the left. What ties [math]\displaystyle{ x }[/math] to [math]\displaystyle{ f(x) }[/math] so tightly? The function itself!

• Graph vs function. Is the graph of a function the function itself? Not quite. The function is a concept, it's not a drawing! The graph is a way to visualise the function to allow us to identity patterns. What can happen in our world is that some objects or shapes may resemble the graph of a function. Trajectories for example. A straight line is just that, a straight line. Now a linear function, the graph of it is a straight line. The line itself is not a function. We often use one or the other interchangeably but under some cases it's good to know the difference.

Concerning trigonometry

• A mistake that can happen is for people to think that when we have a function that is a combination of trig and non-trig functions, the arguments are different. No!! Remember, a radian is just some special case of irrational numbers. There is no need to plug something different in the square root, log or polynomial. I think that a related confusion is because sometimes we read [math]\displaystyle{ \pi }[/math] as [math]\displaystyle{ 1 \pi }[/math], which can cause the confusion between [math]\displaystyle{ \pi }[/math] and one radian.

• When drawing the unit circle, do not do this: [math]\displaystyle{ \cos(x) \times \sin(y) }[/math]. The unit circle is not a function of one variable and even less a function of two variables. It's a circle on the XY Cartesian plane. The axes themselves aren't functions nor angles. If you name the horizontal axis [math]\displaystyle{ \cos(x) }[/math] what you are saying is this "When [math]\displaystyle{ x = 1 }[/math], length is one and [math]\displaystyle{ \cos(1) = 1 }[/math]". Analogously, you are saying this for the vertical axis "When [math]\displaystyle{ y = 1 }[/math], length is one and [math]\displaystyle{ \sin(1) = 1 }[/math]". This mistake means you are, inadvertently, treating sine and cosine as linear functions.

Concerning signs

• Let [math]\displaystyle{ x_1 }[/math] [math]\displaystyle{ x_2 }[/math] be two numbers from the function's domain such that [math]\displaystyle{ x_2 \gt x_1 }[/math]. We cannot conclude that [math]\displaystyle{ f(x_2) \gt f(x_1) }[/math] without knowing whether the function is crescent or not! A similar confusion happens when we think that [math]\displaystyle{ -2 \gt -1 }[/math] because the negative sign inverts the comparison.

• Suppose we have this function [math]\displaystyle{ f(x) = -x }[/math]. If we input [math]\displaystyle{ -x }[/math], the function outputs a positive value, not a negative one! For example: we wish to calculate [math]\displaystyle{ f(-2) }[/math]. Due to some quirk in our minds we confuse the dependent and independent variables and blindly do this [math]\displaystyle{ f(-2) = -2 }[/math].

• [math]\displaystyle{ f^{-1}(x) \neq \frac{1}{f(x)} }[/math]. [math]\displaystyle{ f^{-1} }[/math] means the inverse of the function, which is not the same as calculating the inverse of the value of the function at [math]\displaystyle{ x }[/math]. It's often confusing, but the inverse of a function is a concept. The inverse of a number is another number, usually we think on a fraction. Now a function itself is not a number and the inverse function is also not a number, but the concept that if we can relate [math]\displaystyle{ a }[/math] to [math]\displaystyle{ b }[/math], we can make the reversed relationship, [math]\displaystyle{ b }[/math] to [math]\displaystyle{ a }[/math]. That's how I'd try to differentiate using the same word and notation to mean two different things.
  
  [math]\displaystyle{ -f(x) }[/math] on the other hand means to multiply the function by minus one. This operation essentially mirrors all values of the function at each point. Positive becomes negative. Negative becomes positive. Careful! We are not inverting the sign of the function's argument!

• There is another related confusion that is caused by the words "inverse" and "opposite". Ask somebody what is the inverse / opposite of going up? The person is going to answer that is to go down. In daily life we can interchange both words without causing any harm. In mathematics, however, there is a problem. The inverse of exp is log, but both are functions that are always increasing or decreasing if you invert the sign to negative. There is the confusion! It's not uncommon for people to think that the inverse of a function is a function that is, literally, decreasing if the other is increasing and vice-versa.

Concerning equations

• We often learn that [math]\displaystyle{ x^2 - 1 = 0 }[/math] and [math]\displaystyle{ 2x^2 - 2 = 0 }[/math] have the same roots. When solving equations we can multiply by any constant factor because it doesn't change the roots. However, when plotting the function we cannot do that! To multiply a function by a constant factor changes its graph, hence it's no longer the same function. Think about vectors and linear algebra. When we multiply a vector by a constant, we keep the same orientation and direction but change its magnitude. For example: take the function [math]\displaystyle{ \sin(x) }[/math] or [math]\displaystyle{ \cos(x) }[/math], if we multiply it by a constant, we are changing the function's amplitude but not the points where the graph crosses the x axis.

• When one is learning functions for the first time, the expression [math]\displaystyle{ 1 = 2 }[/math] is never true. However, [math]\displaystyle{ f(1) = 2 }[/math] can be true and that depends on the function. It reads as "function [math]\displaystyle{ f }[/math], calculated at the point [math]\displaystyle{ x = 1 }[/math], is equal to 2" or "the function [math]\displaystyle{ f }[/math] is equal to 2 when we calculate it for [math]\displaystyle{ x = 1 }[/math]".

• Going further in calculus one can find exercises with expressions similar to this: [math]\displaystyle{ f(x^2 + 1) = g(x) }[/math]. Let's call [math]\displaystyle{ h(x) = x^2 + 1 }[/math] then what we have is this: [math]\displaystyle{ f(h(x)) = g(x) }[/math]. Be careful to not confuse it with [math]\displaystyle{ x^2 + 1 = x }[/math] in the same way as above with [math]\displaystyle{ 1 = 2 \ ! }[/math] The equation has functions on both sides. We aren't looking at an equation were the arguments of each function are equal to each other. What the equality is doing is comparing functions.

• Suppose we have [math]\displaystyle{ f(x) = x^2 }[/math] and [math]\displaystyle{ g(x) = x^3 }[/math]. The sum is [math]\displaystyle{ f(x) + g(x) = x^2 + x^3 }[/math]. It can happen that people confuse the argument with the equation like this: [math]\displaystyle{ f(x) + g(x) = (f + g)(x + x) }[/math]. Who said that to sum functions is the same as to add up their respective arguments? A linear transformation looks similar, except that both sides of the equation have the same function! Not two different functions as in here.

• There is a certain type of equation that we don't learn in calculus and it's a functional equation. It's an equation where the unknown is a function itself. Such as [math]\displaystyle{ f(x^2 + 2) = x^4 + 4 }[/math]. If we know that, can we find [math]\displaystyle{ f(x) = \ ? }[/math].

Concerning argument, dependent x independent variables

• Careful with the confusion between a function's argument and the value of the function itself! Let's say we have [math]\displaystyle{ f(x) = x^2 }[/math]. Now what is the difference between [math]\displaystyle{ f(x + 1) = x^2 }[/math] and [math]\displaystyle{ f(x) = (x + 1)^2 \ ? }[/math] In the first case we have a composite function, we have some [math]\displaystyle{ g(x) = x + 1 }[/math] and for every [math]\displaystyle{ x }[/math] that we are going to calculate, first we calculate [math]\displaystyle{ g(x) }[/math], then use the value that [math]\displaystyle{ g(x) }[/math] outputs as the input for [math]\displaystyle{ f(x) }[/math]. We have [math]\displaystyle{ f(g(x)) = x^2 }[/math].
  
  Now for the second case what we have is another function that is not equal to the first. It's pretty obvious that [math]\displaystyle{ x^2 \neq (x + 1)^2 }[/math]. Ergo, these are two different functions. When we are careless and clueless we do this [math]\displaystyle{ f(x + 1) = (x + 1)^2 }[/math]. Who said that [math]\displaystyle{ x = x + 1 \ ?? }[/math] In the previous example, we defined [math]\displaystyle{ g(x) = x + 1 }[/math] and not [math]\displaystyle{ g(x) = x }[/math].
  
  Can we have [math]\displaystyle{ f(x) = g(x)^2 \ ? }[/math] Sure, why not? We can define a function as being the square of a different function. Now, is that a composite function? No. We aren't using one function as the argument of another. What we are doing is saying that the value of the function [math]\displaystyle{ f }[/math], at every point, is going to be the value of the function [math]\displaystyle{ g }[/math], squared.

• This [math]\displaystyle{ \overrightarrow{r}(t) = \lt x(t), \ y(t), \ z(t) \gt }[/math] is not a function of three variables. It's a vector valued function. In this case, it associates time with position in space. In physics, motion in 3D space is not described by just one function. It's one different function for each axis, but one independent variable for all.

• [math]\displaystyle{ \frac{\sin(2x)}{2} \neq \sin \left(\frac{1}{2}2x\right) }[/math]. Sometimes people are tempted to cancel out the two, but we can't just do that! A similar mistake is [math]\displaystyle{ 4 \left( \frac{1}{2} \right)^x \neq 2(1)^x }[/math]. I believe that this is caused by the fact that some functions allow us to do that because of this property [math]\displaystyle{ cf(x) = f(cx) }[/math].