Sets, Relations & Functions
Composition of Functions / Inverse
nta_pyq_2023_jan
Grade 11
Question:
For some $a, b, c \in \mathbb{N}$, let $f(x) = ax - 3$ and $g(x) = x^b + c$, $x \in \mathbb{R}$. If $(f \circ g)^{-1}(x) = \left(\dfrac{x-7}{2}\right)^{1/3}$, then $(f \circ g)(ac) + (g \circ f)(b)$ is equal to ____.
Step-by-Step Solution
Key Concept: Find $a, b, c$ by identifying $fog(x)=2x^3+7$; then compute each composition.
$(fog)^{-1}(x)=\left(\frac{x-7}{2}\right)^{1/3} \Rightarrow fog(x)=2x^3+7 \Rightarrow a=2,b=3,c=5$. $fog(10)=2(10)^3+7=2007$; $gof(3)=gof(b)=(2\cdot3-3)^3+5=27+5=32$. Sum $=2039$.
Correct Answer: 2039