Definition of Square Root Function
A square root function is a type of function that involves square root operations. This function has the general form where is the function inside the square root sign.
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A square root function is a type of function that involves square root operations. This function has the general form where is the function inside the square root sign.
The simplest form of a square root function is . This function takes an input value and produces the square root of that value.
Square root functions have several special characteristics that distinguish them from other functions:
Limited domain: Since the square root of negative numbers is not defined in real numbers, the domain of square root functions is limited to values that make the expression inside the square root non-negative.
Curved graph: The graph of a square root function is a curve that starts from a certain point and continues to rise at an increasingly slower rate.
Always non-negative values: The result of a square root function is always non-negative ().
To understand square root functions well, it's important to determine their domain and range.
The domain of a square root function is all values of that make .
Steps to determine domain:
| Step | Explanation | Example: |
|---|---|---|
| Identify the expression inside the square root | ||
The range of a square root function is all possible output values that the function can produce.
For the function , since square roots always produce non-negative values, then:
Let's visualize the basic square root function .
Square root functions can undergo various transformations that change the shape and position of their graphs.
The function shifts the graph of the basic square root function by to the right (if ) or to the left (if ).
The function shifts the graph of the basic square root function by upward (if ) or downward (if ).
| Function | Transformation | Domain | Range |
|---|---|---|---|
| Basic function | |||
| Shift upward | |||
| Shift downward |
The function with causes vertical dilation on the graph of the square root function.
The general form of a square root function that undergoes transformations is:
Where:
Steps to draw the graph:
| Step | Action | Example: |
|---|---|---|
| Determine starting point | , , Point: |
To solve equations involving square root functions, follow these steps:
| Step | Explanation | Example: |
|---|---|---|
| Isolate the square root | Already isolated | |
| Square both sides |
To solve square root function inequalities, pay attention to the domain and properties of square root functions.
Example: Solve
Combining with the domain condition:
Determine the domain and range of the function
Draw the graph of the function
Solve the equation
A rocket is launched vertically. Its height after is given by . What is the height of the rocket after ?
Determine the value of that satisfies
Domain:
Range:
Drawing the graph
Steps to draw:
| | Identify transformations | (reflection across the -axis), (shift left), (shift up) | | | Determine starting point | , , Starting point: | | | Determine domain | | | | Determine range | Since , the graph decreases from the starting point, so | | | Create value table | Choose values |
Value table:
(Let and )
(from and condition )
| Create the inequality |
| Solve the inequality |
| Write the domain |
| Determine domain |
| Create value table | Choose several values |
| Calculate function values | For : |
| Plot points | Plot , , , etc. |
| Connect points | Create a smooth curve through the points |
| Simplify |
| Solve |
| Verify | ✓ |
| Step | Explanation | Details for |
|---|