Sets, Relations & Functions
Inverse Function — Ratio
nta_pyq_2024_apr
Grade 11
Question:
Let $f(x)=x^5+2x^3+3x+1$, $x\in\mathbb{R}$, and $g(x)$ be a function such that $g(f(x))=x$ for all $x\in\mathbb{R}$. Then $\dfrac{g(7)}{g'(7)}$ is equal to:
Step-by-Step Solution
Key Concept: $f(x)=7$ at $x=1$ (check: $1+2+3+1=7$). $g(7)=1$. $g'(f(x))f'(x)=1\Rightarrow g'(7)=1/f'(1)$. $f'(x)=5x^4+6x^2+3$, $f'(1)=14$.
$g(7)=1$, $g'(7)=1/14$. $g(7)/g'(7)=14$.
Correct Answer: 1