Function and formula

Functions: Domain and range

Function and formula

Take a look at the formula $\orange y=4 \cdot \blue x+2$ .

We can consider this a bit like a machine. If the input of the machine is $\blue x$ , the machine will multiply it by $4$ and then add $2$ to it. The value we then find is the corresponding $\orange y$ -value.

For example $\blue x= \blue 3$ gives $4 \cdot \blue3 +2=14$ , hence, $\orange y=\orange{14}$ .

We can state that the number $\orange {14}$ is the image of the argument $\blue 3$ . Such a "machine" is called a function.

$\begin{array}{lcl} &\blue x \text{ ($\blue{\text{argument}}$)}& \; \\ &\downarrow& \\ &\text{multiplied by }4 & \\ &\downarrow& \\ &\text{ $2$ added}& \\ &\downarrow& \\ &\orange y \text{ ($\orange{\text{image}}$)}& \end{array}$

Fu‍n‍c‍tion

A fu‍nction determines a unique corresponding $\orange{\text{image}}$ for each $\blue{\text{argument}}$ .

Often we can find a corresponding formula with a function.

Example

$\orange y=4 \cdot \blue x +2$

Here, $\blue x$ is the argument, and $\orange{y}$ is the image.

We usually work with functions for which we can write formulas of the form $y=\ldots$ . But this is not always the case, check the example to the right hand side.

Next to that, we also have equations that do not match with a function. The equation of the circle with radius $1$ and center point $\rv{0,0}$ is:
$x^2+y^2=1$

Here, the argument $x=0$ corresponds with two different values of $y$ , $y=1$ or $y=-1$ . But with a function every argument should have a unique image.

Example

The function

$\left\{\begin{array}{ll}y=0 & \text{if } x\lt0 \\ y=1 & \text{if } x \geq 0\end{array}\right.$

is a function, but is of the form $y=\ldots$ .

Dom‍ainSometimes not all values for the argument can be entered in a function.

For example, the function with formula $y=\tfrac{x}{x+3}$ , does not have an image for $x=-3$ .

The arguments of this function are all values except $-3$ . This is called the domain of the function. We will take a closer look at this later.

RangeSometimes, the images of a function are limited.

For example, the function with formula $y=x^2$ , because here the images are always non-negative (since something squared is always non-negative).

The images of this function are all non-negative numbers. This is called the range of a function. We will take a closer look at this later.

Take a look at the formula:
$y=1-x$
Calculate the image of $-1$ ?

The image is: $2$

After all, to calculate the image, we substitute the argument $x=-1$ in the formula
We then get: $y=1-\left(-1\right)=2$
Hence, the image is: $2$ .

New example