If 5 20 what is 12

Calculate function values

In a function, each $$ x $$ value has a $$ y $$ value.

With the function term you can calculate the $$ y $$ values. You sit instead of the variable each a number and then calculate the term.

The $$ y $$ values ​​are also called function values.

Example:
Function: $$ f ($$$$ x $$$$) = 3 $$$$ x $$ $$ - 5 $$

You can calculate the function value for $$ x = $$ $$ 5 $$ as follows:
$$ f ($$$$ 5 $$$$) = 3 * $$ $$ 5 $$ $$ - 5 = 15 $$ $$ - 5 = 10 $$

You can calculate the function value for $$ x = $$ $$ - 1 $$ as follows:
$$ f ($$$$ - 1 $$$$) = 3 * ($$$$ - 1 $$$$) $$ $$ - 5 = $$ $$ - 3 $$ $$ - 5 = $$ $$ - 8 $$

$$ x $$ - value and $$ y $$ - value belong together. They form a pair of values ​​or a point.

You write:
The value pairs $$ (- 1 | -8) $$ and $$ (5 | 10) $$ belong to the function $$ f (x) = 3x-5 $$

Doesn't that look like points in the coordinate system? Correct!

This is how it looks in general:

Function equation:
$$ y = f (x) = mx + b $$ (for each $$ x $$ value)

Function value for $$ x = 2 $$:
$$ f (2) = m * 2 + b $$ (for a certain $$ x $$ value)

Functional term
    ┌─┴──┐
$$ f (x) = 3x-5 $$
└────┬────┘
Function equation

Value pairs and points

As a graph, linear functions always have a straight line.

You can draw the pair of values ​​$$ (x | y) $$ as a point in the coordinate system. The value pairs of the function are the points of the straight lines in the coordinate system.

You can draw the straight line with 2 pairs of values ​​or points.

Example:
After $$ x $$ minutes, the height $$ h (x) $$ of a candle in cm $$ h (x) = $$ $$ - 2/3 x + 20 $$.

To draw the straight line, calculate 2 points that are not too close together.

You reckon:
$$ h (0) = - 2/3 * 0 + 20 = 20 $$ $$ rarr $$ point $$ (0 | 20) $$
$$ h (30) = - 2/3 * 30 + 20 = –20 + 20 = 0 $$ $$ rarr $$ point $$ (30 | 0) $$

$$ x $$ - coordinate
$$ darr $$
Dot $$ ($$$$ 2 $$$$ | $$$$ 3 $$$$) $$
$$ uarr $$
$$ y $$ - coordinate

Here the function is not called $$ f $$, but $$ h $$. Instead of $$ f $$ for any function, one chooses $$ h $$ for the function equation of the height.

The other way around: Calculate $$ x $$ values

It is a bit more difficult when the $$ y $$ is given and you have to calculate the corresponding $$ x $$.

Incidentally, the $$ x $$ values ​​are called arguments.

Example:

Function: $$ f (x) = 3x $$ $$ - 5 $$

What is the name of the $$ x $$ value for the function value $$ 4 $$?

Mathematically: For which $$ x $$ is $$ f (x) = 4 $$?

$$ 3x-5 = 4 $$ $$ | $$ $$ + 5 $$

$$ 3x = 9 $$ $$ | $$ $$: 3 $$

$$ x = 3 $$

The function value $$ y = 4 $$ includes $$ x = 3 $$.

A $$ x $$ value is also called argument or abscissa (from lat. linea abscissa "Cut line")

A $$ y $$ value is also called ordinate (from lat. linea ordinata "Orderly line")

$$ y $$ is dependent on $$ x $$ - as a donkey bridge for the names you can stick to the order in the alphabet:
A before O as well as $$ x $$ before $$ y $$.

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Application task

Anna helps out in the strawberry field during the holidays. She collects the prices for self-picked strawberries.

  • $$ 1 $$ kg strawberries costs $$ 2.50 $$ $$ € $$.
  • Each customer pays an additional $$ 0.50 $$ $$ € $$ to allow them to nibble a little while picking.

Anna writes down the functional equation $$ y = f (x) = 2.5 * x + 0.5 $$ and calculates different pairs of values.

Example 1:
How much do $$ 2 $$ kg of picked strawberries cost?
$$ y = f (2) = 2.5 * 2 + 0.5 = 5.5 $$

$$ 2 $$ kg of picked strawberries cost $$ 5.50 $$ $$ € $$.

Example 2:
Mr. Lu pays $$ 13.00 $$ $$ £ $$. How many kg of strawberries did he pick?

$$ y = f (x) = 13.00 $$

$$ 2.5 * x + 0.5 = 13.00 $$ $$ | $$ $$ - 0.5 $$

$$ 2.5 * x = 12.50 $$ $$ | $$ $$: 2.5 $$

$$ x = 5 $$

Mr. Lu picked $$ 5 $$ kg of strawberries.

Table of values

So that Anna doesn't have to calculate every time, she has created a table of values:

$$ y = f (x) = 2.5 * x + 0.5 $$

Weight in kg ($$ x $$)Price in euros ($$ y $$)
$$1,0$$
$$3,00$$
$$1,5$$
$$4,25$$
$$2,0$$
$$5,50$$
$$2,5$$
$$6,75$$
$$3,0$$
$$8,00$$
$$3,5$$
$$9,25$$
$$4,0$$
$$10,50$$
$$4,5$$
$$11,75$$
$$5,0$$
$$13,00$$

The graph for this:


A Table of values is clear if you more than 2 Calculate points of the graph.

Tip calculator:

Some pocket calculators do the arithmetic for a table of values ​​for you - take a look at the instructions for use!

A bit of theory at the end

Domain of definition

The domain is all numbers that you can insert into a function, i.e. all $$ x $$ values.

For linear functions: $$ D = QQ $$

Range of values

The definition range are function values ​​($$ y $$ values) that can be obtained when calculating the function term.

For linear but not constant functions: $$ W = QQ $$

$$ QQ $$ are the rational numbers: all positive and negative fractions.

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