https://file.aiquickdraw.com/mathbot/file/1725448318933zmomijyt.jpg

The function given in the image is \( f(x) = \sqrt{2x - 6} \). The task is to find the domain and range of this function.

### Step 1: Finding the Domain

The domain of a function is the set of all possible input values (i.e., \( x \) values) for which the function is defined. Since \( f(x) \) involves a square root, the expression inside the square root must be non-negative (greater than or equal to zero) because the square root of a negative number is not defined in the set of real numbers.

The expression inside the square root is \( 2x - 6 \). To ensure this expression is non-negative, set up the inequality:

\[
2x - 6 \geq 0
\]

### Step 2: Solving the Inequality

Now, solve the inequality \( 2x - 6 \geq 0 \):

1. **Add 6 to both sides** to isolate the term with \( x \):

\[
2x \geq 6
\]

2. **Divide both sides by 2** to solve for \( x \):

\[
x \geq 3
\]

This inequality shows that \( x \) must be greater than or equal to 3. Therefore, the domain of the function is all real numbers \( x \) such that \( x \geq 3 \).

### Domain Conclusion:
\[
\text{Domain} = [3, \infty)
\]
This means \( x \) can be any value from 3 to infinity.

### Step 3: Finding the Range

The range of a function is the set of all possible output values (i.e., \( f(x) \) values) the function can produce.

For the function \( f(x) = \sqrt{2x - 6} \):

1. Since the square root function always produces non-negative values (i.e., \( \sqrt{y} \geq 0 \) for any \( y \geq 0 \)), \( f(x) \) must be greater than or equal to 0.

2. To find the minimum value of \( f(x) \), consider the smallest \( x \) in the domain. The smallest \( x \) value is 3 (from the domain). Substituting \( x = 3 \) into the function gives:

\[
f(3) = \sqrt{2(3) - 6} = \sqrt{6 - 6} = \sqrt{0} = 0
\]

Thus, the minimum value of \( f(x) \) is 0.

3. As \( x \) increases, the value inside the square root \( (2x - 6) \) also increases, and so does the value of \( f(x) \). As \( x \) approaches infinity, \( f(x) \) will also approach infinity. 

### Range Conclusion:
\[
\text{Range} = [0, \infty)
\]
This means \( f(x) \) can be any value from 0 to infinity.

### Summary

- **Domain:** \( [3, \infty) \)
- **Range:** \( [0, \infty) \)

Would you like more details or have any questions?

---

Here are five related questions you might explore:

1. How would the domain and range change if the function were \( f(x) = \sqrt{2x + 6} \)?
2. What happens to the domain and range if the function is \( f(x) = \sqrt{-2x + 6} \)?
3. How can we determine the domain and range of a quadratic function?
4. What are the effects of shifting or scaling the graph of \( f(x) = \sqrt{2x - 6} \)?
5. How would you find the domain and range of a function involving a cube root?

**Tip:** When finding the domain of functions involving square roots, always ensure the expression under the square root is non-negative.

Find the domain and range of the following function: f(x) = √(2x − 6)

Learn how to find the domain and range of the function f(x) = √(2x − 6) with a detailed step-by-step solution. This problem covers key algebraic concepts such as inequalities and square root properties, making it suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation