Determine the domain of the function and express in interval notation, f(x)=square root of -7 -x

The function given is \( f(x) = \sqrt{-7 - x} \).

### To determine the domain of this function:
1. The square root function is defined for non-negative values (i.e., the expression inside the square root must be greater than or equal to zero).
   
2. Therefore, we need:
   \[
   -7 - x \geq 0
   \]
   Solving this inequality:
   \[
   -x \geq 7
   \]
   Multiply both sides by \(-1\), which reverses the inequality:
   \[
   x \leq -7
   \]

### Domain:
The domain of the function is all values of \( x \) such that \( x \leq -7 \).

### Expressing in interval notation:
The domain is \( (-\infty, -7] \).

Would you like more details or have any questions?

Here are some related questions:
1. What happens to the domain if the function is \( f(x) = \sqrt{x + 7} \)?
2. How does the domain change for \( f(x) = \sqrt{-7 - 2x} \)?
3. What is the domain of \( f(x) = \sqrt{x^2 - 4x - 12} \)?
4. How can we graph the function \( f(x) = \sqrt{-7 - x} \)?
5. What is the range of the function \( f(x) = \sqrt{-7 - x} \)?

**Tip:** Always remember that square roots of negative numbers are undefined in the real number system.

Determine the domain of the function and express in interval notation, f(x) = square root of -7 - x.

This problem demonstrates how to find the domain of the function f(x) = √(-7 - x) by solving an inequality. Learn to express the domain in interval notation, covering key concepts such as square root functions and inequalities. Suitable for Grades 9-11.

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