English: R code to reproduce the image:

``` options(

 repr.plot.width  = 10,   # in inches (default = 7)
 repr.plot.height = 13    # in inches (default = 7)

)

library(ggplot2) library(gridExtra)

1. Define the functions

linear_function <- function(x) x concave_function <- function(x) sqrt(x) convex_function <- function(x) x^2 identity_line <- function(x) x

1. Generate data

x_values <- seq(0.001, 10, length.out = 1000) data_linear <- data.frame(x = x_values, y = linear_function(x_values)) data_concave <- data.frame(x = x_values, y = concave_function(x_values)) data_convex <- data.frame(x = x_values, y = convex_function(x_values))

1. Create plots

plot_linear <- ggplot(data_linear, aes(x, y)) +

 geom_line() +
 geom_abline(intercept = 0, slope = 1, linetype = "dashed", color = "red") +
 ggtitle("Linear: y = x") + ylim(0,10) + xlim(0,10)

plot_concave <- ggplot(data_concave, aes(x, y)) +

 geom_line() +
 geom_abline(intercept = 0, slope = 1, linetype = "dashed", color = "red") +
 ggtitle("Concave: y = x^(1/2)") + ylim(0,10) + xlim(0,10)

plot_convex <- ggplot(data_convex, aes(x, y)) +

 geom_line() +
 geom_abline(intercept = 0, slope = 1, linetype = "dashed", color = "red") +
 ggtitle("Convex: y = x^2") + ylim(0,10) + xlim(0,10)

1. Log10 transformation

data_linear_log <- data.frame(x = log10(x_values), y = log10(linear_function(x_values))) data_concave_log <- data.frame(x = log10(x_values), y = log10(concave_function(x_values))) data_convex_log <- data.frame(x = log10(x_values), y = log10(convex_function(x_values)))

plot_linear_log <- ggplot(data_linear_log, aes(x, y)) +

 geom_line() +
 geom_abline(intercept = 0, slope = 1, linetype = "dashed", color = "red") +
 ggtitle("Linear (Log10-Log10): y = x") + ylim(0,1) + xlim(0,1)

plot_concave_log <- ggplot(data_concave_log, aes(x, y)) +

 geom_line() +
 geom_abline(intercept = 0, slope = 1, linetype = "dashed", color = "red") +
 ggtitle("Concave (Log10-Log10): y = x^(1/2)") + ylim(0,1) + xlim(0,1)

plot_convex_log <- ggplot(data_convex_log, aes(x, y)) +

 geom_line() +
 geom_abline(intercept = 0, slope = 1, linetype = "dashed", color = "red") +
 ggtitle("Convex (Log10-Log10): y = x^2") + ylim(0,1) + xlim(0,1)

1. Arrange the plots in a grid

grid.arrange(plot_linear, plot_linear_log,

            plot_concave, plot_concave_log, 
            plot_convex, plot_convex_log, 
            ncol = 2, top = "Comparison of Linear, Concave, and Convex Functions\nIn original (left) and log10 (right) scales")

```

The R code provided above generates a grid of six plots, arranged in two columns and three rows, showcasing different mathematical functions both in their original scale and in a log10-log10 scale. The functions displayed include a linear function (y = x), a concave function (y = sqrt(x)), and a convex function (y = x^2). Each of these functions is depicted in two plots: one showing the relationship in its original scale and the other showing the relationship in a log10-log10 scale.

In each plot: - The data is represented by a black line. - A dashed red line is included in each plot, representing a line with a 0 intercept and a slope of 1, for reference. - The x-axis and y-axis are labeled appropriately, with the axes in the log10-log10 plots labeled as log10(x) and log10(y). - Each plot has a title describing the function and indicating whether it is in the original scale or in the log10-log10 scale.

This visualization serves to demonstrate how different types of mathematical relationships are represented both in their normal scale and when transformed into a logarithmic scale.