# 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 ($$ y $$) |
---|---|

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.

*kapiert.de*can do more:

- interactive exercises

and tests - individual classwork trainer
- Learning manager

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