find the domain and the range of this graph: f(x)= -x

The function you provided is \( f(x) = -x \), which is a linear function with a slope of -1. Let's go through the steps to find its domain and range.

### **Domain:**
The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the function \( f(x) = -x \), there are no restrictions on \( x \)—you can input any real number. Thus, the domain is:

\[
\boxed{\mathbb{R}} \quad \text{(all real numbers)}
\]

### **Range:**
The range of a function is the set of all possible output values (y-values). Since the function is linear and spans all real numbers, for every input \( x \), there is a corresponding output \( y = -x \). This means the function can take any real value for \( y \). Therefore, the range is also:

\[
\boxed{\mathbb{R}} \quad \text{(all real numbers)}
\]

### **Summary:**
- **Domain:** \( \mathbb{R} \) (all real numbers)
- **Range:** \( \mathbb{R} \) (all real numbers)

Would you like more details on linear functions or the properties of the graph? Or do you have any other questions?

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Here are 5 related questions you might find interesting:

1. How does the slope of a linear function affect the graph's domain and range?
2. What happens to the domain and range of a function if it is quadratic instead of linear?
3. How do you find the domain and range for piecewise functions?
4. Can a function have a restricted domain or range? If so, how?
5. What are some examples of functions with specific domains and ranges, like exponential or trigonometric functions?

**Tip:** Remember, for any linear function like \( f(x) = mx + b \), the domain and range are both typically \( \mathbb{R} \), unless specific constraints are imposed.

Discover the domain and range of the linear function f(x) = -x. This problem explores how the function spans all real numbers, making its domain and range both equal to the set of all real numbers. A great learning resource for Grades 7-9 students.

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