What is the domain and range of the function y equals square root of 4 x plus 12 end root?

To find the domain and range of the function \( y = \sqrt{4x + 12} \), let's analyze it step by step.

### Domain:
The domain of a function is the set of all possible values of \( x \) for which the function is defined. For the function \( y = \sqrt{4x + 12} \), the expression inside the square root, \( 4x + 12 \), must be non-negative because the square root of a negative number is not a real number.

So, we need to solve the inequality:
\[
4x + 12 \geq 0
\]

Subtract 12 from both sides:
\[
4x \geq -12
\]

Divide both sides by 4:
\[
x \geq -3
\]

Thus, the domain of the function is:
\[
\text{Domain}: \, x \geq -3
\]
In interval notation, this is:
\[
\text{Domain}: \, [-3, \infty)
\]

### Range:
The range of a function is the set of all possible values of \( y \) that the function can take. Since \( y = \sqrt{4x + 12} \), the output \( y \) represents the square root of a non-negative number.

The square root function always produces non-negative values (i.e., \( y \geq 0 \)). The smallest value of \( y \) occurs when \( 4x + 12 = 0 \), which happens when \( x = -3 \). In this case:
\[
y = \sqrt{4(-3) + 12} = \sqrt{0} = 0
\]
As \( x \) increases beyond \(-3\), the value of \( 4x + 12 \) increases, making \( y \) increase as well. There is no upper bound on \( y \) as \( x \) increases.

Thus, the range of the function is:
\[
\text{Range}: \, y \geq 0
\]
In interval notation, this is:
\[
\text{Range}: \, [0, \infty)
\]

### Summary:
- **Domain**: \( x \geq -3 \) or \([-3, \infty)\)
- **Range**: \( y \geq 0 \) or \([0, \infty)\)

Would you like more details or have any questions?

### Relative Questions:
1. How do you find the domain of a function involving a square root?
2. What happens to the domain if the function is \( y = \sqrt{4x - 12} \) instead?
3. How do you determine the range of a quadratic function?
4. What would the graph of the function \( y = \sqrt{4x + 12} \) look like?
5. How does the domain and range change if the function was \( y = -\sqrt{4x + 12} \)?

### Tip:
When dealing with square root functions, always check the expression inside the square root to determine the domain.

Learn how to find the domain and range of the function y = sqrt(4x + 12) step-by-step with clear explanations and examples. Understand the concepts of functions, square roots, inequalities, and interval notation. Suitable for high school students.

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