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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.
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\[\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} \]
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A function determines a unique corresponding #\orange{\text{image}}# for each #\blue{\text{argument}}#.
Often we can find a corresponding formula with a function.
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Example
#\orange y=4 \cdot \blue x +2#
In here #\blue x# is the argument and #\orange{y}# the image.
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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 equation 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#.
Sometimes 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.
Sometimes 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:
\[y=9\cdot x^3-9\cdot x^2-3\cdot x+9\]
Calculate the image of #-2#?
The image is: #-93#
After all, to calculate the image, we substitute the argument #x=-2# in the formula
We then get: \[y=9\cdot \left(-2\right)^3-9\cdot \left(-2\right)^2+\left(-3\right)
\cdot \left(-2\right)+9=-93\]
Hence, the image is: #-93#.
Function and formula
