To find the range of the function $f(x) = 54x - 13$ over the domain $-2 < x < 20$:

Steps:

1. The function $f(x) = 54x - 13$ is a linear function, which means it is continuous and strictly increasing (since $54 > 0$) over its domain.

2. Evaluate $f(x)$ at the endpoints of the domain:

   • Lower endpoint ($x = -2$): $f(-2) = 54(-2) - 13 = -108 - 13 = -121$
   • Upper endpoint ($x = 20$): $f(20) = 54(20) - 13 = 1080 - 13 = 1067$

3. Since $f(x)$ is strictly increasing, the range is: $\boxed{(-121, 1067)}$

This range excludes $-121$ and $1067$ because the domain excludes $x = -2$ and $x = 20$. If the endpoints were included, the range would be $[-121, 1067]$.

Would you like more detailed explanations or further assistance?

Related Questions:

1. What is the slope of the function $f(x)$ and how does it affect the range?
2. How would the range change if $x$ were a closed interval, $[-2, 20]$?
3. What are the intercepts of $f(x) = 54x - 13$?
4. How would you find the inverse of $f(x) = 54x - 13$?
5. What does the steepness of the line $54x - 13$ tell us about its behavior over this interval?

Tip: For linear functions, always check the values at the endpoints of the domain to determine the range!