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2 problems match “counting distinct values”. Clear

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1. Distinct values of f(x) minus x ●○○○○ 10 ptsSuppose the function $f(x)$ is defined explicitly by the table $$\begin{array}{c || c | c | c | c | c} x & 0 & 1 & 2 & 3 & 4 \\ \hline f(x) & 0 & 0 & 1 & 3 & 6 \end{array}$$ This function is defined only for the values of $x$ listed in the table. Suppose $g(x)$ is defined as $f(x)-x$ for all numbers $x$ in the domain of $f$. How many distinct numbers are in the range of $g(x)$?
2. Possible resistances from four resistors ●●●○○ 50 ptsRebecca has four resistors, each with resistance 1 ohm. Every minute, she chooses any two resistors with resistances $a$ and $b$ ohms, and combines them into one by one of the following methods: (i) connect them in series, producing a resistor of $a+b$ ohms; (ii) connect them in parallel, producing a resistor of $\frac{ab}{a+b}$ ohms; (iii) short-circuit one of the two resistors, producing a resistor of either $a$ or $b$ ohms. Suppose that after three minutes, Rebecca has a single resistor with resistance $R$ ohms. How many possible values are there for $R$?