Exercise 12.1 - Problem 10

Give the domain and range of the multi-variable function.

\(f(x, y) = \frac{1}{x + 2y}\)

f <- function(x, y) {
  1 / (x + 2 * y)
}

x <- seq(-10, 10, by = 1)
y <- seq(-10, 10, by = 1)

grid <- expand.grid(x = x, y = y)

grid$f_values <- with(grid, f(x, y))

val_values <- grid[!is.nan(grid$f_values), ]

f_range <- range(val_values$f_values)

cat("Range of valid function values:\n")
## Range of valid function values:
print(f_range)
## [1]  -1 Inf

Range of valid function values = -1 Inf indicates that the function \(f(x, y) = \frac{1}{x + 2y}\) can take on any value from negative infinity to positive infinity, except zero.

The key Takeaways from the 605 Course:
The main point of the course was to learn how different mathematical fields such as probability, calculus, and linear regression come together to support data science. One of my favorite examples from the course was the Gambler’s Ruin problem. This topic showed how probability helps us understand a gambler’s likelihood of losing all their money versus winning more, demonstrating the practical use of math in real-life scenarios.