find the range and domain of f(x)= 2x+x / x

The function you provided is \( f(x) = \frac{2x + x}{x} \). We can simplify this function before determining the domain and range.

### Step 1: Simplify the function
\[
f(x) = \frac{2x + x}{x} = \frac{3x}{x} = 3 \quad \text{(for } x \neq 0\text{)}.
\]
Thus, \( f(x) = 3 \) for all \( x \neq 0 \).

### Step 2: Domain
The domain of a function is the set of all possible inputs (values of \( x \)) for which the function is defined. Since the function \( f(x) = \frac{3x}{x} \) has \( x \) in the denominator, the only value that makes the denominator zero is \( x = 0 \). Therefore, \( f(x) \) is undefined at \( x = 0 \).

**Domain:** \( x \in (-\infty, 0) \cup (0, \infty) \).

### Step 3: Range
The range of a function is the set of all possible outputs (values of \( f(x) \)). Since the function simplifies to \( f(x) = 3 \) for all \( x \neq 0 \), the output is always 3, except for \( x = 0 \), where the function is undefined.

**Range:** \( \{ 3 \} \).

### Final Answer:
- **Domain:** \( (-\infty, 0) \cup (0, \infty) \)
- **Range:** \( \{ 3 \} \)

Would you like further details or have any questions?

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Here are 5 related questions for practice:
1. What is the domain and range of \( f(x) = \frac{1}{x} \)?
2. How does the domain of \( f(x) = \frac{x^2 + x}{x} \) differ from your original function?
3. If \( f(x) = \frac{2x+1}{x-1} \), what is its domain and range?
4. What happens if you multiply both numerator and denominator by \( x^2 \)?
5. How does removing the denominator affect the domain of a rational function?

**Tip:** Always check for values that make the denominator zero to find restrictions on the domain of rational functions.

Find the range and domain of f(x) = 2x + x / x.

Discover how to find the domain and range of the function f(x) = (2x + x) / x, where x ≠ 0. This problem focuses on simplifying rational functions and determining the restrictions on the domain and range, suitable for students in Grades 9-11.

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