how to find the value of x

I don't know if it is useful and practical for you, but my idea is to fit a (polynomial) curve to your data and then use this curve to "predict" (find) your x value for any y value. In case your y value corresponds to multiple x values you'll keep the first one.

I suggest you to run the process step by step to see how your initial dataset gets transformed.

          library(ggplot2) library(dplyr)  # The time variable time <- seq(from = 0, to = 120, by=10) # The measured variable value <- c(5, 5.5, 7.8, 8.3, 9.5, 10.9, 11.5, 12, 13, 14, 12.5, 11.1, 9) # Put together df <- data.frame(time, value)  # Plot value (x axis) againt time (y axis) ggplot(df, aes(time, value)) +    geom_point()

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You need a process that excludes the overlapping parts. I'm using a process that spots when the "value" value (x axis) starts getting smaller. Those cases are excluded.

          # create a row index df %>% mutate(id = row_number()) -> df  df_updated =      df %>%     group_by(id) %>%          # for each row     do(data.frame(.,max_value = max(df$value[df$id <= .$id]))) %>%   # obtain the maximum value up to that point     ungroup() %>%     filter(value >= max_value)     # exclude declining parts   # Plot value (x axis) againt time (y axis) from the updated dataset ggplot(df_updated, aes(time, value)) +    geom_point()

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Those are the data points you need to consider.

          # filt a polynomial curve that best describes your data fit <- lm(time~poly(value,8,raw=TRUE), data = df_updated)   ## NOTE that here it requires some extra work to find which degree gives you an acceptable fit (you can create a process that calculates your optimal degree. Here I used 8).  # check how good your fitting is ggplot(df_updated, aes(time, value)) +    geom_point() +   geom_line(aes(predict(fit, df_updated), value))

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          # get the time at value = 10 predict(fit, data.frame(value=10))   #        1   # 41.67011