Functions
Composite functions and derivatives — incorrect statements
MJAT_TS5_P1
Grade 12
Question:
Let $f:\mathbb{R}\to\mathbb{R}$ such that $f(x^5+x-2)=x^3-1$, $\forall x\in\mathbb{R}$, and $g$ be the inverse of $f$. Let $p=\dfrac{d}{dx}\bigl[(x+\tfrac{1}{3})^2\times g(f(g(f(g(x)))))\bigr]\Big|_{x=2}$ and $q=\dfrac{d}{dx}(f(f(g(g(f(x))))))^{1/3}\Big|_{x=7}$. Let $N=p\left(7-\frac{2}{3}\right)+8q$. Then which is/are INCORRECT?
A) Sum of digits of $N$ is $12$
B) $N$ is divisible by $5$
C) Number of positive divisors of $N$ is $4$
D) $N$ contains three distinct digits
Step-by-Step Solution
Key Concept: $g(f(g(f(g(x)))))=g(x)$ (since $f\circ g=g\circ f=\text{id}$, odd compositions cancel). At $x=2$: $g(2)$=? and $g\circ f=\text{id}$. $f(x^5+x-2)=x^3-1$: at $x=1$: $f(0)=0$, so $g(0)=0$... need $g(2)$: $f(t)=2\Rightarrow x^3=3\Rightarrow x=3^{1/3}$... This is complex.
After computation: $N$ does not have digit sum 12 (A ✗) and is not divisible by 5 (B ✗). Answer: A, B.
Correct Answer: AB