The following WitCH is pretty old, but it came up in a tutorial yesterday, so what the Hell. (It’s also a good warm-up for another WitCH, to appear in the next day or so.) It comes from the 2011 Mathematical Methods Exam 1:
For part (a), the Examination Report indicates that f(g)(x) =√([x+2][x+8]), leading to c = 2 and d = 8, or vice versa. The Report indicates that three quarters of students scored 2/2, “However, many [students] did not state a value for c and d”.
For Part (b), the Report indicates that 84% of students scored 0/2. After indicating the intended answer, (-∞,-8) U (-2,∞) (-∞,-8] U [-2,∞) or R
(-8,-2), the Report goes on to comment:
“This question was very poorly done. Common incorrect responses included [-3,3] (the domain of f(x); x ≥ -2 (as the ‘intersection’ of x ≥ -8 with x ≥ -2); or x ≥ -8 (as the ‘union’ of x ≥ -8 with x ≥ -2). Those who attempted to use the properties of composite functions tended to get confused. Students needed to look for a domain that would make the square root function work.”
The Report does not indicate how students got “confused”, although the composition of functions is briefly discussed in the Study Design (page 72).
Maximal domain of which function ….? f, g or f(g)? I think you could argue a case for the alternative answer -3 leq x leq 3 (the maximal domain of f for which f(g(x)) is defined) ….
If the answer is R(-8,-2), shouldn’t the alternative notation have square brackets? Or am I just being as pathetically pedantic as VCAA?
Thanks, RF. You’re absolutely correct, but that was my transcription error, not a VCAA error. Fixed now.
Also, surely x at least -2 is the maximal domain of the function once it is composed if we ignore how the function was composed? I’ve often wondered (not out loud, that would be causing trouble) why it matters how the function was composed and why this should have an effect on the resulting maximal domain.
I will refer this query to any of the superior intellects (meant seriously) who frequent this site.
Hi RF.
I’m not sure what you’re asking here ….
Assuming that f is defined over its maximal domain (an assumption VCAA has made and apparently expects students to make), then the maximal domain of f(g(x)) will be the maximal domain of whatever the function f(g(x) turns out to be – how f(g(x) was composed can be ignored.
However, this is not the case if f is not defined over its maximal domain. For example, let’s say that the domain for f in the above question was given as [0, 3]. Then the requirement ran(g) subset dom(f) must be carefully applied:
Only -5 leq x leq -2 permits g(x) = x + 5 to have the range [0, 3] and so the maximal domain of f(g(x)) is -5 leq x leq -2.
Clearly how f(g(x) was composed cannot be ignored in this case. Note that this method can be applied when f does have its maximal domain of [-3, 3]:
-8 leq x leq -2 permits g(x) = x + 5 to have the range [-3, 3] and so the maximal domain of f(g(x)) is -8 leq x leq -2.
I would argue that in many ways this is a simpler and more natural way of answering the question (but the nudge in part (a) shows that VCAA clearly had other ideas).
(By studying this problem, you can probably tell that I didn’t attend RMIT: https://mathematicalcrap.com/2019/11/06/posww-8-easy-solutions/#comment-1475)
JF, assuming f and g have their maximal domains, you say that “the maximal domain of f(g(x)) will be the maximal domain of whatever the function f(g(x)) turns out to be”. I don’t think that agrees with the Study Design.
And yet I assume the raison d’être of part(a) is setting up this method ….?
I was a bit rash in saying that how f(g(x) was composed can be ignored when f and g are defined over their maximal domains. For example, if f(x) = 1/x and g(x) = 1/(x – 3) are both defined over their maximal domains, then f(g(x)) = x – 3 but the restriction x neq 3 must be carried through to get this simplification so obviously how it was composed is clearly important. Otherwise one would think that the maximal domain of f(g(x)) was R.
This actually answers my question, so thanks. If f(g(x))=x-3 then one would be forgiven for assuming the maximal domain is R. The fact that the function wandered around in C^2 space for a bit before settling back in R^2 space is lost without the details of how it was composed. Still don’t think VCAA study design has it 100% correct though somehow (not sure why, just a feeling)
RF, see SRK’s comment below.
Yeah, I wasn’t reading the question properly… read it as a simple square-root function, not the root of a quadratic. Makes a bit of a difference!
I think a large part of the problem is how this is commonly taught (perhaps not by readers of this site) / how it appears in the popular textbooks – VCAA’s culpability lies in not clarifying the issue in the study design. In the textbooks we are told that f(g(x)) can be defined only if the range of g(x) is a subset of the domain of f(x). In this question, we are not given domains for g and f, so I guess we should take the implied range of g(x) to be R, and the implied domain of f(x) to be [–3, 3], and thus the confusion for students.
Thank, SRK. This is exactly the point, but it is in the Study Design as well as the textbooks. If the students were “confused” it is because the idiots at the VCAA confused them.
I am feeling very confused here. Is not the implied domain for f(x) R(-3,3)?
Yes Damo you are quite right. For whatever silly reason I read f(x) as root(9 – x^2).
Echoing Marty: if you were “confused” it is because the idiot making the comment above confused you.
Just a couple suggestions for those commenting on this WitCH. First, it turns out wordpress hates backslashes. So, I suggest you use a minus sign dash or whatever. Secondly, to the substance of the post, many of you are trying to use common sense, which may not be a winning strategy. After all, this is a WitCH. I suggest you begin with the study design, and work from there.