find the the domain and range of functions
1. y= √2x +4

Let's find the domain and range of the function \( y = \sqrt{2x + 4} \).

### 1. **Domain**:

The domain of a function refers to all the possible values of \( x \) that can be plugged into the function without making it undefined. Since we have a square root function, the expression inside the square root must be non-negative (i.e., \( \geq 0 \)), because the square root of a negative number is not defined in the real numbers.

Thus, we solve:
\[
2x + 4 \geq 0
\]
\[
2x \geq -4
\]
\[
x \geq -2
\]
So, the domain of the function is:
\[
\boxed{x \geq -2 \text{ or } [-2, \infty)}
\]

### 2. **Range**:

The range of a function refers to all the possible values of \( y \) that can result from plugging in the values of \( x \) from the domain.

Since \( y = \sqrt{2x + 4} \), the square root function always produces non-negative outputs. The minimum value of \( 2x + 4 \) occurs when \( x = -2 \), and in that case:
\[
y = \sqrt{2(-2) + 4} = \sqrt{0} = 0
\]
As \( x \) increases beyond -2, \( y \) increases as well without any upper bound.

Thus, the range of the function is:
\[
\boxed{y \geq 0 \text{ or } [0, \infty)}
\]

### Summary:
- **Domain**: \( [-2, \infty) \)
- **Range**: \( [0, \infty) \)

Would you like more details on any step?

#### 5 Related Questions:
1. How do we find the domain of a rational function?
2. What happens to the range if the function is \( y = \sqrt{-2x + 4} \)?
3. How do we calculate the domain and range of a logarithmic function?
4. Can you explain how to find the domain and range of trigonometric functions?
5. What changes occur in the domain and range if the function involves an absolute value?

#### Tip:
When dealing with square roots, always ensure the expression inside the root is non-negative to determine the domain.

find the domain and range of functions y = √2x + 4

Learn how to find the domain and range of the function y = √(2x + 4) with a detailed step-by-step solution. This problem covers key concepts such as the square root function, inequality solving, and determining the domain and range. Suitable for students in Grades 9-11.

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