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-# Functions
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+# Functions
+
+## Functions of a Single Variable
+
+A function describes the relationship between variables.
+
+Examples:
+
+$$f(x) = x^2$$
+
+$$y = 2+3\cdot x^4$$
+
+![Fig. 4](../media/5_1_Functions_of_a_single_variable.png)
+
+### Details
+
+Functions are commonly used in statistical applications, to describe relationships.
+
+:::note Definition
+
+A **function** describes the relationship between variables.
+A variable $y$ is described as a function of a variable $x$ by completely specifying how $y$ can be computed for any given value of $x$.
+
+:::
+
+An example could be the relationship between a dose level and the response to the dose.
+
+The relationship is commonly expressed by writing either $f(x) = x^{2}$ or $y = x^2$.
+
+Usually names are given to functions, i.e. to the relationship itself.
+For example, $f$ might be the function and $f(x)$ could be its value for a given number $x$.
+Typically $f(x)$ is a number but $f$ is the function, but the sloppy phrase "the function $f(x)=2x+4$" is also common.
+
+### Examples
+
+:::info Example
+
+$f(x) = x^2$ or $y = x^2$ specifies that the computed value of $y$ should always be $x^2$, for any given value of $x$.
+
+:::
+
+## Functions in R
+
+A function can be defined in R using the `function` command:
+
+![Fig. 5](../media/5_2_Functions_in_R.png)
+
+## Ranges and Plots in R
+
+Functions in R can commonly accept a range of values and will return a corresponding vector with the outcome.
+
+### Examples
+
+:::info Example
+
+```text
+f <- function(x) {return(x*12)}
+x <- seq (-5,5,0,1)
+y <- f(x)
+plot {(x,y) type= 'l'}
+```
+
+:::
+
+## Plotting Functions
+
+In statistics, the function of interest is commonly called the response function.
+If we write $y=f(x)$, the outcome $y$ is usually called the response variable and $x$ is the explanatory variable.
+Function values are plotted on vertical axis while $x$ values are plotted on horizontal axis.
+This plots $y$ against $x$.
+
+### Examples
+
+:::info Example
+
+The following R commands can be used to generate a plot for function $y= 2+3x$:
+
+```text
+x <- seq(0:10)
+g <- function(x) {
++ yhat <- 2+3*x
++ return(yhat)
++ }
+
+x <- seq(0,10,0.1)
+y <- g(x)
+plot(x,y,type="l", xlab="x",ylab="y")
+```
+
+:::
+
+## Functions of Several Variables
+
+### Examples
+
+:::info Example
+
+$$\begin{aligned} z &= 2x+3y+4\\ v &= t^2+3x\\ w &= t^2+3b \cdot x\end{aligned}$$
+
+:::