A Level Maths : Common Mistakes/misconceptions Watch

4D Chess
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NotNotBatman
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Transformation of graphs such as ln(4-x).

First define a function f(x) =ln(x)
Then apply a transformation f(-x) = ln(-x)
Define a new function g(x) = ln (-x)

DO NOT WRITE g(x+4)
You Must put brackets around ALL terms when replacing x.
g(x+4) = ln(-(x+4)) =ln(-x-4) is incredibly wrong and is a common error.

Instead it is g(x-4) =ln(-(x-4)) =ln(4-x)

A different transformation.
Last edited by NotNotBatman; 1 month ago
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NotNotBatman
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There's a lack of understanding of how functions work. There is a function called the identity function which as the name suggests "doesnt change anything"

So ff^{-1}(x) = id(x) =x
Looks confusing but it isnt. Say you need to solve gf = a. But we only have a sketch of f.
Then all you have to do is introduce the identity function by post pre multiplying (in the sense of functions) both sides g inverse to obtain g^-1gf=g^-1(a) but g(g^-1) is the identity function and doesn't change the function it acts on, so we get f=g^-1(a)

Now if we only have a graoh of f where we can work out described values and we have an equation for g, we can now work it out.
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NotNotBatman
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#64
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Clasic mistake, You cannot cancel over addition
one is  \dfrac{2x}{2+x} YOU CANNOT CANCEL THE 2s here!
You can only cancel over multiplication with common factors. To make sure you can cancel try factorising first, even if it seems easy.
Last edited by NotNotBatman; 1 month ago
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NotNotBatman
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For those of you dealing with complex numbers, remember for pure imaginary numbers sqrt(ab) = sqrt(a)sqrt(b) is generally NOT true.

Easy example: Sqrt[(-1)(-1)] =1
Sqrt(-1) *sqrt(-1) = -1
Not equal

the above rule works for non negative a and b only.
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NotNotBatman
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One mistake i find is taking "success" in binomial distribution to be its English meaning. For example the probability of failing can be your "success" probability.

A "success" is only an event for which there exists and opposite event and not a literal English definition of the word.

You might get a question where tbe probability of an event which in literal english is a failure is p, but in the mathematical random variable sense this is your success.

Confusion can be avoided if you forget succuss and failure all together and just remember there are 2 outcomes.
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_gcx
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(Original post by NotNotBatman)
Confusion of notation.
Thinking that  \dfrac{\mathrm{d}y}{\mathrm{d}x} is different from \frac{\mathrm{d}}{\mathrm{d}x} (y)

Saying things such as" find the dy/dx of y=... "
"Find the differential of ..."
These seem nitpicky, but it may cause some to write things down that don't make any sense.
Some also say derive for differentiate probably confused by the term derivative. Differentiate being the preferred modern term in English, derive being used in some other languages.
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Glaz
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#68
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Not really a mistake but putting this for everyone because it will come up

DO NOT FORGET THE +C
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_gcx
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Something that's not required at A-level but useful to know for conceptual understanding: The question of "what's the domain" of a function is nonsense - the domain is an integral part of defining a function.

The functions f:[0,1] \to \mathbb R and g:\mathbb R \to \mathbb R with f(x) = x for all x \in [0,1] and g(x) = x for all x \in \mathbb R are considered different functions, and either [0,1] or \mathbb R are equally valid domains to choose.

What a question is usually asking is "what's the largest subset of \mathbb R for which this rule defines a function". Which is basically asking where the expression is well-defined.

Another linguistic note on that note: f(x) isn't the function, f is, f(x) is the function f evaluated at some point x. So we shouldn't talk about "the function f(x)" we should say "the function f".
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NotNotBatman
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(Original post by _gcx)
Something that's not required at A-level but useful to know for conceptual understanding: The question of "what's the domain" of a function is nonsense - the domain is an integral part of defining a function.

The functions f:[0,1] \to \mathbb R and g:\mathbb R \to \mathbb R with f(x) = x for all x \in [0,1] and g(x) = x for all x \in \mathbb R are considered different functions, and either [0,1] or \mathbb R are equally valid domains to choose.

What a question is usually asking is "what's the largest subset of \mathbb R for which this rule defines a function". Which is basically asking where the expression is well-defined.

Another linguistic note on that note: f(x) isn't the function, f is, f(x) is the function f evaluated at some point x. So we shouldn't talk about "the function f(x)" we should say "the function f".
It is widely accepted that if a domain or codomain isn't specified, then we are talking about the natural domain. So i would say asking what's the domain actually does make sense, but is just shorthand as many things happen to be.
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_gcx
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(Original post by NotNotBatman)
It is widely accepted that if a domain or codomain isn't specified, then we are talking about the natural domain. So i would say asking what's the domain actually does make sense, but is just shorthand as many things happen to be.
Yeah I suppose - I still feel like it's a thing worth clarifying since it doesn't seem to be clear to everyone.
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Sir Cumference
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#72
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I’m not a moderator but I think it would be helpful if the notation was kept to A Level. This thread is meant to be helpful for A Level students, not undergraduates.
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IrrationalRoot
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On the topic of functions, A Level students seem to not actually understand what a function actually is sometimes. I've seen many cases where they try to justify that e.g. f defined by f(x)=0 is a function because you can write it as f(x)=x \times 0. Of course the variable x need not appear explicitly in the definition of f(x) - a function a simply a rule assigning to each input a single output. In this case, the rule is just 'for any input, f outputs  0 .'

(Original post by _gcx)
Another linguistic note on that note: f(x) isn't the function, f is, f(x) is the function f evaluated at some point x. So we shouldn't talk about "the function f(x)" we should say "the function f".
Although this is technically correct, in practice you will see the function referred to as f(x) very often at any level of mathematics, because it is frequently useful to make it easier to refer to the argument of f or make it clear what the standard letter for the argument of that function is. E.g. 'The Riemann zeta function \zeta(s) is defined for \Re (s)>0 by ...'. Nonetheless it is still a common misconception that f(x) is actually the function itself so I agree that this should be clarified and understood beforehand (as with any abuse of notation).
Last edited by IrrationalRoot; 1 month ago
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IrrationalRoot
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(Original post by RDKGames)
\sqrt{a^2 + b^2} \neq a+b

:facepalm:
What? Pythagoras' Theorem isn't a+b=c?!
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NotNotBatman
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(Original post by Sir Cumference)
I’m not a moderator but I think it would be helpful if the notation was kept to A Level. This thread is meant to be helpful for A Level students, not undergraduates.
I was thinking about this when i used id(x).

Thing is you do have to know about how the identity function works if i recall correctly, but you don't exactly have to know its notation.

Im wondering if that's okay? Because i didnt want to say ff^(-1) magically cancels out the next term. It would be analogous to saying e and ln cancel out without teaching what inverse functions are, but i do want to be aware of the audience, because i appreciate that after a few years of undergraduate its easy to forget and get carried away, which im trying to avoid.
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NotNotBatman
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Not so much a common mistake, but a mistake which can easily be avoided.

Draw the graph of arcsin(x)

Instead of thinking "oh god which one is it? Is it up to heaven or down to hell, if i get it wrong, ill wake up in the middle of ths night when im 35 thinking about that moment where I needed an A in maths and I got a B with 1 matk off"

You can instead just try putting a few values in your calculator. Starting with the end values. If you're really unsure don't forget a table of values as a last resort to see how the shape of the graph goes. If asked to draw the graph of y=arcsinx, you can think i know the general shape of inverse trig but i cant remember which is which, then find arcsin(1) and arcsin(-1) now you know 2 points the graph goes thru and now you can match it to what's in your memory bank.
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IrrationalRoot
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Thinking that 'inverse trig functions' are actually inverses of the trig functions defined on \mathbb{R}. It's important to understand that \sin and \cos defined on \mathbb{R} are not invertible because they are 2\pi-periodic, but they are when the domain is restricted to a suitable interval (and this should be understood geometrically). And that consequently, \arcsin(\sin x) is not necessarily equal to x. This can be quite a tricky concept for A Level students to fully understand but it's a great test of whether they understand how inverse functions work.
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any of these for y12 students?
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Sir Cumference
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(Original post by NotNotBatman)
Transformation of graphs such as ln(4-x).

First define a function f(x) =ln(x)
Then apply a transformation f(-x) = ln(-x)
Define a new function g(x) = ln (-x)

DO NOT WRITE g(x+4)
You Must put brackets around ALL terms when replacing x.
g(x+4) = ln(-(x+4)) =ln(-x-4) is incredibly wrong and is a common error.

Instead it is g(x-4) =ln(-(x-4)) =ln(4-x)

A different transformation.
This is a very common mistake. Whenever I see a question like that in the exam I know it’s going to be done very badly.
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NotNotBatman
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(Original post by Sir Cumference)
This is a very common mistake. Whenever I see a question like that in the exam I know it’s going to be done very badly.
I made a thread on it myself a few years ago!
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