# What is a function in mathematics 5

## Features basics

On this page on the topic of "Basics of Functions" you will find explanations on the following topics:

**Table of Contents**

### Linear function

The general form for a linear function is:

\ begin {align *}

y = m \ cdot x + b \ quad \ textrm {with} \ quad m = \ frac {y_2-y_1} {x_2-x_1}

\ end {align *}

To determine the slope $ m $ we need two points $ P_1 (x_1 | y_1) $ and $ P_2 (x_2 | y_2) $.

**Here you can find free tutorial videos on the subject of linear functions.**

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### Quadratic functions

The general form for a quadratic function is:

\ begin {align *}

y = ax ^ 2 + bx + c

\ end {align *}

The simplest quadratic function is the normal parabola with $ y = x ^ 2 $. The highest or lowest point of a quadratic function is also called the vertex $ S $. The vertex shape is:

\ begin {align *}

y = a \ cdot (x- \ varvec {d}) ^ 2 + e \ quad \ textrm {with} \ quad S (\ varvec {d} | e)

\ end {align *}

**Complete square functions playlist**

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### Polynomial function

The general form for a polynomial function (also called a rational function)

**3rd degree reads:**

$ y = ax ^ 3 + bx ^ 2 + cx + d $

**4th degree reads: **

$ y = ax ^ 4 + bx ^ 3 + cx ^ 2 + dx + e $

Degree $ n $ describes the highest exponent for $ x $ for $ a \ neq 0 $. There are at most as many zeros as the degree $ n $ of the function.

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### Root function

The general form of a root function is:

\ begin {align}

f (x) = \ sqrt [n] {x} \ \ text {for} \ x \ geq 0 \ notag

\ end {align}

with $ n $ as the root exponent. It has the only zero at $ x = 0 $. The larger $ n $, the flatter the graph from $ x = 1 $. If $ n $ is even or odd, $ x \ in [0, \ infty) $ or $ x \ in \ mathbb {R} $.

### Amount function

In mathematics, the absolute value function assigns a real number its distance from zero. The so-called absolute amount, absolute value or simply amount, is always a non-negative number, i.e. greater than or equal to zero.

**Spellings:** $ f (x) = | x | $ or $ f (x) = abs (x) $.

For any real number $ x $, the following applies:

\ begin {align *}

| x | = \ left \ {\ begin {array} {ll} x &, \ x \ geq 0 \

-x &, \ x <0 \ end {array} \ right. .

\ end {align *}

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### Exponential function

A function is called an exponential function (to base a) if it has the form

\ begin {align}

f (x) = a ^ x = e ^ {x \ cdot \ ln (a)} \ \ text {with} \ x \ in \ mathbb {R}, a> 0 \ notag

\ end {align}

where $ a $ denotes any positive constant. If $ a = e $, one generally speaks of the $ e $ function. This is Euler's number $ e \ approx 2.72 $ - an irrational number such as the circle number $ \ pi $. It runs above the $ x $ axis and has no zero.

The form of the exponential function reminds us of a power expression, whereby the role of base and exponent is reversed! In the event that $ a = e $, the following applies as a consequence of the power laws for the $ e $ function: $ e ^ 0 = 1, \ e ^ 1 = e, \ e ^ x \ cdot e ^ y = e ^ {x + y} $.

\ begin {align *}

e ^ 0 = 1, \ \ e ^ 1 = e, \ \ e ^ x \ cdot e ^ y = e ^ {x + y}

\ end {align *}

### Logarithmic function

A function is called a logarithm function (base $ a $) if it has the general form

\ begin {align *}

f (x) = \ log_a (x), \ x \ in (0, \ infty)

\ end {align *}

where $ a $ denotes any positive constant.

In the special cases $ a = e $ and $ a = 10 $ one speaks of

- $ f (x) = \ ln (x) $, as "natural logarithm" and
- $ f (x) = \ log (x) $, as "decadic logarithm".

As a rule, however, we calculate with the natural logarithm. If, however, the case should arise that there is no natural logarithm, this can be rewritten using simple means as follows:

\ begin {align *}

log_a (x) = \ frac {\ ln (x)} {\ ln (a)}

\ end {align *}

Another useful link is

\ begin {align *}

e ^ {\ ln (x)} = x \ \ textrm {or} \ \ ln (e ^ x) = x,

\ end {align *}

which is extremely important in the field of "solving equations".

**Logarithmic Laws**

\ begin {align *}

& \ ln (ab) = \ ln (a) + \ ln (b) \

& \ ln (\ frac {a} {b}) = \ ln (a) - \ ln (b) \

& \ ln (a ^ b) = b \ cdot \ ln (a)

\ end {align *}

### Manipulation of basic functions

Also called graph transformation. Idea: From the graph of a given function $ f (x) $ with the definition range $ D $ and the value range $ W $, the graphs of "new" functions $ g (x) $ with the definition range $ D_g $ and the value range $ W_g $ can be gained through simple operations.

Here is an overview table that describes the manipulation of functions and the effect on the graph, the definition range and the value range. "Effect" should mean: If the term $ g (x) $ is formed as described, the graph of $ g $ arises from the graph of $ f $ through ...

$

\ begin {array} {c | c | c | c}

g (x) = & D_g = & W_g = & \ text {Effect on the graph} \

\ hline \ hline

f (x) + a, \ a \ in \ mathbb {R} & D & a + W & \ text {vertical shift by a} \

\ hline

f (x + a), \ a \ in \ mathbb {R} & -a + D & W & \ text {horizontal shift by -a} \

\ hline

c \ cdot f (x), \ c> 0 & D & c \ cdot W & c> 1: \ text {stretching, 0

f (c \ cdot x), \ c> 0 & \ frac {1} {c} \ cdot D & W & c> 1: \ text {compression}, 0

-f (x) & D & -W & \ text {reflection on y-axis} \

\ hline

-f (x) & D_f & -W_f & \ text {mirroring on x-axis}

\ end {array}

$

**Change within the function, e.g. $ f (x-a) $ $ \ stackrel {\ wedge} {=} $ Horizontal manipulation**

- The domain of definition changes
- The range of values remains the same

**Change outside the function, e.g. $ f (x) + a $ $ \ stackrel {\ wedge} {=} $ Vertical manipulation**

- The domain of definition remains the same
- The range of values changes

In the following we will present the most common manipulations or transformations using an example. The normal parabola serves as the output function

\ begin {align *}

f (x) = x ^ 2, \ quad x \ in \ mathbb {R}.

\ end {align *}

### Shift in the x direction

We can easily take the shift in the $ x $ direction into account in our function equation. To do this, we first take a look at the graphs in the following coordinate system.

The vertex of this parabola and all other points have been shifted 2 units to the right, starting from the normal parabola. If we take a look at the function equation, we see that it reads as follows:

\ begin {align *}

g (x) = (x-2) ^ 2

\ end {align *}

A shift in the $ x $ direction can always be recognized by the fact that the value by which the function was shifted appears with the opposite sign in brackets.

We also want to look at a parabola that should be shifted to the left.

The functional equation of this parabola is:

\ begin {align *}

g (x) = (x + 2) ^ 2

\ end {align *}

The parabola has been shifted 2 units to the left. In general we can say: The function $ f (x-a) $ shifts by $ + a $ along the $ x $ axis.

### Shift in y-direction

We recognize the shift in the $ y $ direction from the fact that the value by which the function was shifted in the $ y $ direction is appended with the correct sign without parentheses.

Starting from the normal parabola, we consider the following functions:

\ begin {align *}

g (x) = x ^ 2-2 \ quad \ quad \ textrm {and} \ quad \ quad h (x) = x ^ 2 + 2

\ end {align *}

For this we look at the following coordinate system.

The $ -2 $ in the equation of $ g (x) $ means that the normal parabola is shifted down 2 units. Similarly, the $ + 2 $ results in a shift of 2 units upwards. In general we can say: The function $ f (x) + a $ shifts by $ + a $ along the $ y $ axis.

Of course, it is also possible to carry out both a shift in the $ x $ direction and a shift in the $ y $ direction at the same time. To do this, we consider the parabola in the following figure.

The functional equation is:

\ begin {align *}

g (x) = (x-2) ^ 2-2

\ end {align *}

In the bracket we can see the shift by 2 units to the right and behind the bracket we see the shift by 2 units down. A functional equation in the form is called a vertex form. This makes it possible to read off the coordinates of the vertex directly. In our case $ S (2 | -2) $.

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### Compression and elongation

If we want to stretch or compress a function, we have to multiply the function by a factor $ c $. In our example with the normal parabola, $ f (x) = x ^ 2 $ then becomes $ g (x) = c \ cdot f (x) = c \ cdot x ^ 2 $. The following applies to the factor $ c $ if

\ begin {align *}

c> 1 \ quad & \ Rightarrow \ & \ textrm {stretching} \

0

The factor $ c $ indicates whether it is a stretching or a compression and is either directly in front of the $ x ^ 2 $ or, if our function equation should be in the vertex form, directly in front of the bracket.

The normal parabola $ x ^ 2 $ has the factor $ c = 1 $. However, for reasons of mathematical laziness, we do not write these down. The normal parabola is neither stretched nor compressed.

A straight parabola could be the equation

\ begin {align *}

g (x) = 2 \ cdot x ^ 2

\ end {align *}

to have.

Since the prefactor $ c $ is greater than 1, the parabola is stretched. The graph is much narrower than the normal parabola. Each $ y $ value is multiplied by the factor $ c = 2 $.

If the factor $ c $ is between 0 and 1, the function is compressed

\ begin {align *}

g (x) & = 0.5 \ cdot x ^ 2

\ end {align *}

describes a compressed normal parabola which is wider than the normal parabola $ f (x) = x ^ 2 $. Each $ y $ value is multiplied by the factor $ c = 0.5 $.

In general we can say: The function $ c \ cdot f (x) $ is stretched if $ c> 1 $ and compressed if $ 0

A distinction can be made between vertical and horizontal compression or stretching. What we just got to know was vertical compression and stretching.

We can formulate the horizontal one in general as follows: The function $ f (c \ cdot x) $ is stretched if $ 0

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### reflection

We recognize a function mirrored on the $ x $ axis by the fact that there is a minus in front of the function. In our example, the normal parabola mirrored on the $ x $ axis is

\ begin {align *}

g (x) = - f (x) = - x ^ 2.

\ end {align *}

The graph of $ g $ and $ f $ is shown in the figure below. Each $ y $ value is multiplied by $ (- 1) $.

As with compression and stretching, we can divide this into vertical and horizontal mirroring. However, the horizontal mirroring, i.e. the mirroring on the $ y $ axis, is only possible to a limited extent here. Because the mirrored function $ g (x) = f (-x) = (- x) ^ 2 = x ^ 2 = f (x) $ is nothing else than the normal parabola itself. Why? Because the square turns the minus into a plus.

If we consider the function $ f (x) = x ^ 3 $ and would like to mirror this on the $ y $ axis, the transformed function reads

\ begin {align *}

g (x) = f (-x) = (- x) ^ 3 = -x ^ 3

\ end {align *}

Each $ x $ value is multiplied by $ (- 1) $. Since the function is point symmetrical, the horizontal reflection is equal to the vertical. In general we can say: The function $ -f (x) $ or $ f (-x) $ is mirrored on the $ x $ or $ y $ axis.

**another example** (not so detailed) In the following we will give a small example sketch for each operation from the table. The function serves as the output function

\ begin {align *}

f (x) = \ sqrt {x}

\ end {align *}

with the definition range $ x \ in D: = [0; 4] $ and the value range $ W = [0; 2] $.

For the parameters $ a $ and $ c $ we use $ a = 5 $ and $ c = 2 $ as an example! We will see how the graphs of the functions change and also indicate the new definition and value range ($ D_g $, $ W_g $). We see the graph of the output function in the adjacent figure.

**Horizontal shift:** $ g (x) = f (x + a) = f (x + 5) = \ sqrt {x + 5} $

**Horizontal reflection:** $ g (x) = f (- x) = \ sqrt {-x} $

**Horizontal compression**

$ g (x) = f (c \ cdot x) = f (2 \ cdot x) = \ sqrt {2x} $

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**Vertical stretch:**

$ g (x) = c \ cdot f (x) = 2 \ cdot f (x) = 2 \ cdot \ sqrt {x} $

**Vertical shift:**

$ g (x) = f (x) +5 = \ sqrt {x} +5 $

**Vertical mirroring:**

$ g (x) = - f (x) = - \ sqrt {x} $

**Check out Daniel's Features Playlist for more helpful videos.**

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### Inverse function

For a function $ f (x) $, $ f ^ {- 1} (x) $ is an inverse function if for $ y = f (x) $: $ x = f ^ {- 1} (y) $. So if you insert a function value $ y $ of the output function into the inverse function, you get the corresponding $ x $ value.

**Action**:

- Rewrite the function as $ y = f (x) $ and solve it step by step for $ x $.
- Swap variables $ x $ and $ y $.
- Write down the inverse function $ f ^ {- 1} (x) $ or $ \ bar {f} (x) $.

A function can be reversed if each function value $ y $ is only assumed in one place $ x \ in D_f $: $ f (x_1) = f (x_2) \ Rightarrow x_1 = x_2 $.

Such a function $ f $ has an inverse function $ f ^ {- 1} $, defined by $ f ^ {- 1} (y) = x $ for $ y = f (x) $, i.e. $ f ^ {- 1 } \ left (f (x) \ right) = x $. In some cases the domain of a function has to be restricted so that the restricted function can be reversed.

### Examples

**a)**Linear function: Find the inverse function of $ f (x) = 2x + 1 $.

We work through the above procedure and solve the equation for $ x $.

\ begin {align *}

& \ & y \ & = 2x + 1 \ quad && | -1 \

& \ Leftrightarrow \ & y-1 \ & = 2x \ quad && |: 2 \

& \ Leftrightarrow \ & 0.5y-0.5 \ & = x \ quad &&

\ end {align *}

Exchanging $ x $ and $ y $ to $ y = 0.5x-0.5 $ yields the inverse function

\ begin {align *}

f ^ {- 1} (x) = 0.5x-0.5.

\ end {align *}

Graphical representation of the function and its inverse function:

**b)**Quadratic function: Find the inverse function of $ f (x) = (x + 2) ^ 2 $.

We work through the above procedure and solve the equation for $ x $.

\ begin {align *}

& \ quad & y \ & = (x + 2) ^ 2 \ quad && | \ sqrt {} \

& \ Leftrightarrow \ quad & \ pm \ sqrt {y} \ & = x + 2 \ quad && | -2 \

& \ Leftrightarrow \ quad & \ pm \ sqrt {y} -2 \ & = x \ quad &&

\ end {align *}

Swapping $ x $ and $ y $ to $ y = \ pm \ sqrt {x} -2 $ yields the inverse functions

\ begin {align *}

f ^ {- 1} _1 (x) & = \ sqrt {x} -2 \ \ textrm {and} \ f ^ {- 1} _2 (x) & = - \ sqrt {x} -2.

\ end {align *}

Graphical representation of the function and its inverse function:

A problem arises with a quadratic function such as $ y = x ^ 2 $. There is no clear assignment here, because two $ x $ values are assigned to one $ y $ value. An inverse function can then be defined for a part, as in the example of the normal parabola with $ f ^ {- 1} _1 (x) = \ sqrt {x} $ for the positive part and $ f ^ {- 1} _2 (x ) = - \ sqrt {x} $ for the negative part.

**Note**: Not every function has a corresponding inverse function.

**For a deeper understanding, you should watch Daniel's tutorial video on the subject of inverse functions!**

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### What is given in the function?

In application tasks we have to understand what the function actually describes. Often it is about filling levels of some reservoirs or speeds of aircraft.It is therefore very important to know what, for example, the derivation of the speed means in a factual context.

The following overview should serve as a summary. If the following is specified for $ f (t) $ in our function, then is

**Check out Daniels' tutorial on how to use functions!**

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**For more information, check out Daniel's Feature Playlist!**

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