# If x is equal to sqrt x 2

### The square root function $$ y = sqrt (x) $$

You already know the roots. There is also a new function variety! That too. Here we go:

Each area x of a square has a clearly defined side length y with the assignment: **Area x $$ rarr $$ side length y**.

The area of the square is: $$ y ^ 2 = x $$.

So: You calculate the side length from the area with $$ y = sqrt x $$.

Table of values for this assignment:

x | 0 | 0,16 | 0,64 | 1 | 4 | 9 |
---|---|---|---|---|---|---|

y | 0 | 0,4 | 0,8 | 1 | 2 | 3 |

**The root function**

Function equation: $$ y = f (x) = sqrt (x) $$

Domain of definition of f: $$ RR ^ (ge0) $$ (real numbers greater than or equal to 0)

Value range of f: $$ RR ^ (ge0) $$

Designation: square root function or root function for short

### The square root function as an inverse function

The extraction of the roots is the reverse of the squaring. The square function is $$ y = f (x) = x ^ 2 $$.

Will the **Domain of definition** the square function $$ y = f (x) = x ^ 2 $$ on the area $$ x ge 0 $$ **limited**, there is exactly one x-value associated with each y-value. The function $$ f $$ thus has an inverse function $$ f ^ -1 $$.

### Computational determination of the inverse function

**Step 1:** Solving y = f (x) for x:

$$ x ^ 2 = y = f (x) | sqrt () $$

$$ x = sqrt (y) $$**2nd step:** Swap the variables:

$$ y = sqrt (x) $$**3rd step:** Note the inverse function:

$$ f ^ -1 (x) = sqrt (x) $$

The inverse function $$ f ^ -1 $$ is the **Root function**.

The graph of the root function goes through **reflection** the **Square function** on the straight line y = x.

The square function $$ f (x) = x ^ 2 $$ with $$ xge 0 $$ and the root function $$ f ^ -1 (x) = sqrt (x) $$ are inverse functions to each other.

The term under the root is called **Radicand**. He is allowed to **not negative** become.

### Shift of the square root function I.

Additional functions can be formed by adding the root term of the root function. Compare the root function with the shifted root function. **Example 1:**

Properties:

- Function equation: $$ y = sqrt (x) + 3 $$
- Definition range: $$ RR ^ (ge0) $$
- Value range: $$ RR ^ (ge3) $$
**vertical**Shift around**3**Units according to**above**

**Example 2:**

Properties:

- Function equation: $$ y = sqrt (x - 2) $$
- Definition range: $$ RR ^ (ge2) $$
- Value range: $$ RR ^ (ge0) $$
**horizontal**Shift around**2**Units according to**right**

HORIZONTAL = HORIZONTAL = $$ larr $$$$ rarr $$

V S

E E

R N

T K $$ uarr $$

I = R =

K E $$ darr $$

A C

L H

T

*kapiert.de*can do more:

- interactive exercises

and tests - individual classwork trainer
- Learning manager

### Combined

You can also move the function in both directions.

**Example 3:**

Properties:

- Function equation: $$ y = sqrt (x + 1) - 2 $$
- Definition range: $$ RR ^ (ge-1) $$
- Value range: $$ RR ^ (ge-2) $$
**vertical**Shift around**2**Units according to**below****horizontal**Shift around**1**Unit after**Left**

### Stretching or compressing the root function

**Example 1:**

Properties:

- Function equation: $$ y = 3 sqrt (x -2) - 3 $$
- Definition range: $$ RR ^ (ge2) $$
- Value range: $$ RR ^ (ge-3) $$
**vertical**Shift around**3**Units according to**below****horizontal**Shift around**2**Units according to**right****Elongation**with the factor**3**

**Example 2:**

Properties:

- Function equation: $$ y = -0.5 sqrt (x - 1) $$
- Definition range: $$ RR ^ (ge1) $$
- Value range: $$ RR ^ (le0) $$
**vertical**Shift around**0**units**horizontal**Shift around**1**Unit after**right****Upsetting**with the factor**0,5****reflection**on the x-axis

### generalization

With the help of the parameters a, b and c in the function equation $$ y = a sqrt (x - b) + c $$ you can shift, stretch or compress the root function $$ y = sqrt (x) $$.**Vertical stretch** or **Upsetting** to the **Factor a**,

For **a <0** is on the x-axis **mirrored****Horizontal shift**g around **b units**,

For **b> 0** to **right** and

For **b <0** to **Left****Vertical shift** around **c units**n,

For **c <0** to **below** and for **c> 0** to **above**.

*kapiert.de*can do more:

- interactive exercises

and tests - individual classwork trainer
- Learning manager

- What makes a video game profitable
- Can I get Indian food abroad
- Who is the author of An Autobiography
- Who is the poorest billionaire
- What are the basics of American culture
- Adds vape water to the lungs
- How do you follow leads on Facebook
- Do you remember all of your teachers
- Can games be played like
- How do I send money to India
- Why do people love junk food
- Why do non-Americans hate Americans
- What is the ideal habitat for hyacinth macaws
- How did God speak in Genesis
- Why do all countries block work visas
- Runs low testosterone levels
- What makes a person really happy?
- Is Thailand a high income economy
- What are some good innovative ideas
- Amazon delivers after 9 p.m.
- Which unanswered questions annoy you
- How effective are purchase funnels
- What are the best binary options webinars
- Are cocoapods widely used in iOS development