1. **State the problem:** We need to find the rule for the function $i(x)$ given a table of bounded linear functions with some known properties for $f(x)$, $g(x)$, $h(x)$, and unknown for $i(x)$. 2. **Recall the general form of a linear function:** $$ y = mx + b $$ where $m$ is the rate of change (slope) and $b$ is the initial value (y-intercept). 3. **Analyze the given functions:** - $f(x)$ has rate of change $m=1$ and passes through $(2, -1)$. - $g(x)$ has zero at $x = -\frac{12}{5}$ and initial value $b=2$. - $h(x)$ has domain $[-15, 18]$, range $[-3, 19]$, and is increasing. - $i(x)$ is unknown. 4. **Use the information about $h(x)$ to find its slope:** Since $h(x)$ is increasing and linear, slope $m = \frac{\text{change in } y}{\text{change in } x} = \frac{19 - (-3)}{18 - (-15)} = \frac{22}{33} = \frac{2}{3}$. 5. **Find $h(x)$'s equation:** Use point-slope form with point $(-15, -3)$: $$ y - (-3) = \frac{2}{3}(x - (-15)) \\ y + 3 = \frac{2}{3}(x + 15) \\ y = \frac{2}{3}x + 10 - 3 = \frac{2}{3}x + 7 $$ 6. **For $i(x)$, since no information is given, we can infer it might be a constant function or zero slope function bounded similarly to others.** 7. **Assuming $i(x)$ is constant, its rule is:** $$ i(x) = c $$ where $c$ is a constant value. 8. **If more information is provided, such as domain or range, we could specify $c$. Without it, the best we can say is $i(x)$ is a constant function.** **Final answer:** The rule for $i(x)$ is a constant function: $$ i(x) = c $$ where $c$ is a constant.