The function $g$ is such that $g(x) = 3x^2  12$ for $x <q$, find the greatest possible value of $q$.
Thanks to Mirko, now we have a complete statement of Q10:
The function $f$ is such that $f(x)=2x+3$ for $x\geq 0$. The function $g$ is such that $g(x)=ax^2+b$ for $x\leq q$, where $a$, $b$ and $q$ are constants. The function $f\circ g$ is such that $(f\circ g)(x)=6x^2−21$ for $x\leq q$. (i) Find the values of $a$ and $b$. (ii) Find the greatest possible value of $q$. [2] It is now given that $q=3$. (iii) Find the range of $f\circ g$. (iv) Find an expression for $(f\circ g)^{1}(x)$ and state the domain of $(f\circ g)^{1}$.
OP solved Q10(i) and found that $g(x)=3x^212$.
Q10(ii) is asking the greatest possible value of $q$ such that $f(g(x))=6x^221$ is defined for $x\leq q$ (here $f(x)=2x+3$ is defined for $x\geq 0$). Therefore the composition is possible if $g(x)\geq 0$. Hence we should find the greatest possible value of $q$ such that $g(x)\geq 0$ FOR ALL $x\leq q$.
By solving the inequality we get that $g(x)\geq 0$ iff $x\in (\infty,2]\cup [2,+\infty)$. So what is $q$?
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user450199
Updated on December 20, 2022Comments

user450199 11 months
This question is from CIE A Level Maths (9709) May/June 2016 Q10ii. The answer is $q = 2$, but I don't get how it is done. Anyone, please help me and explain to me. Thank you.

Hagen von Eitzen over 6 yearsNothing prevents $g(x)=3x^212$ for all $x\in \Bbb R$, so $q$ is not bounded in any way

DMcMor over 6 yearsSurely there is more to the question than that.

JB1 over 6 yearsIs there any more information given? It seems like more information is necessary for the solution.

fleablood over 6 yearsThe question makes no sense. You can set q to anything and say the function is that for any value of x less than q. There is no restriction or criterion. The question simply makes no sense.

Toby Mak over 6 yearsThere are so many past papers to search for in that year and season. Why don't you specify what paper it is?

Mirko over 6 yearscould you find a link to your question? There are too many "May/June 2016 Q10ii" one of them is as 8mundo.com/file?id=3353147 and seems to come close to the statement of your question, but does not seem to be exactly your question. There is a lon glist at 8mundo.com/post/…

Robert Z over 6 years@Mirko I think that you found the right question paper!

Mirko over 6 yearsThe correct statement is now a comment to the correct answer ... of course the OP could have provided it, and formatted it properly, hopefully this will happen in their future posts


Mirko over 6 yearsyou mean OP found $g(x)=3x^212$, not $g(x)=3x^2+12$. I see now what they mean in that paper, indeed it is the right paper, where by $fg(x)$ they mean composition, $(f\circ g)(x)$, it makes sense now. (And of course $q=2$ the OP already knew that :)

Robert Z over 6 years@ Mirko yes, thanks

Mirko over 6 yearsTo make it selfcontained, here is the statement. The function $f$ is such that $f(x)=2x+3$ for $x\ge0$. The function $g$ is such that $g(x)=ax^2+b$ for $x\le q$, where $a,b$ and $q$ are constants. The function $fg$ is such that $fg(x) = 6x^2−21$ for $x\le q$. (i) Find the values of $a$ and $b$. [3], (ii) Find the greatest possible value of $q$. [2], It is now given that $q=−3$. (iii) Find the range of $fg$. [1], (iv) Find an expression for $(fg)^{−1}(x)$ and state the domain of $(fg)^{−1}$, [3].