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

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

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

2017 UNM-PNM Contest II (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
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2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 3

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