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.