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

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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 shiftg around b units,
For b> 0 to right and
For b <0 to Left

Vertical shift around c unitsn,
For c <0 to below and for c> 0 to above.

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