2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 3

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Problem

Let $f(x)=\frac{x}{x-1}$ and $g(x)=\frac{x}{3x-1}$.

(a) Determine $f\circ g(x)$ and $g\circ f(x)$.

(b) Denote $\underbrace{h\circ h\circ \cdot \circ h}_\text{n times} := h^n$. Determine all the functions in the set $S = \{H | H = (g\circ f)^n\circ g$ or $H = (f\circ g)^n\circ f$ for some $n$ a whole number$\}$.

Solution

$S=\Big\{ \frac{x}{(2n+1)x-1} | n\in \mathbb{Z} \Big \}$

See also

2017 UNM-PNM Contest II (ProblemsAnswer KeyResources)
Preceded by
Problem 2
Followed by
Problem 4
1 2 3 4 5 6 7 8 9 10
All UNM-PNM Problems and Solutions
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