what are the formula of domain and range of Radical function 

A radical function typically involves the square root (or other roots) of a variable. The most common form is a **square root function**, which can be written as:

\[
f(x) = \sqrt{g(x)}
\]

### 1. **Domain of a Radical Function**
For square root functions, the expression inside the square root (called the *radicand*) must be **non-negative** because square roots of negative numbers are not real.

- **Domain Formula**: 
  \[
  g(x) \geq 0
  \]

To find the domain:
- Set the radicand \( g(x) \) ≥ 0 and solve for \( x \).
- The solution gives the set of all values of \( x \) for which the function is defined.

### 2. **Range of a Radical Function**
The range depends on the values that the radical function can produce.

- **For the square root function \( f(x) = \sqrt{g(x)} \)**:
  \[
  f(x) \geq 0
  \]

To find the range:
- Consider the minimum and maximum values that the function can output. For most square root functions, the output is non-negative (since the square root of any non-negative number is non-negative).
  
For example, if \( g(x) \geq 0 \), then \( f(x) = \sqrt{g(x)} \geq 0 \).

### Examples:

1. **For \( f(x) = \sqrt{x-3} \)**:
   - **Domain**: Set \( x-3 \geq 0 \) → \( x \geq 3 \).
   - **Range**: \( f(x) \geq 0 \), so the range is \( [0, \infty) \).

2. **For \( f(x) = \sqrt{-(x^2 - 4)} \)**:
   - **Domain**: Set \( -(x^2 - 4) \geq 0 \) → \( x^2 \leq 4 \) → \( -2 \leq x \leq 2 \).
   - **Range**: Since \( f(x) \geq 0 \), the range is still \( [0, \infty) \).

### Key Points:
- The domain depends on making the radicand non-negative.
- The range for square root functions is typically \( [0, \infty) \), unless the function has been shifted or transformed.

---

Would you like a detailed example or further clarifications?

Here are 5 related questions:
1. How do you find the domain of a cube root function?
2. What happens to the domain when you add or subtract a number inside the radical?
3. How do transformations affect the range of a radical function?
4. How do you determine the domain and range of higher-order root functions (like cube roots)?
5. Can a radical function have a range that includes negative values?

**Tip**: When working with square roots, always check for restrictions on the domain before attempting to graph or solve the function.

what are the formula of domain and range of Radical function

Explore the domain and range of radical functions, focusing on square roots. Learn how to determine the domain using the inequality g(x) ≥ 0 and find the range with f(x) ≥ 0. Suitable for students in Grades 9-12, with examples included.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation